Lobachevskii Journal of Mathematics Vol. 12, 2003, 11 – 39 \begin{matrix}c \\ ◯ \end{matrix} A.N. Chuprunov, O.V. Rusakov

A.N. Chuprunov, O.V. Rusakov

CONVERGENCE FOR STEP LINE PROCESSES UNDER SUMMATION OF RANDOM INDICATORS AND MODELS OF

MARKET PRICING

(submitted by D. Kh. Mustari)

Abstract. Functional limit theorems for random step lines and random broken lines defined by sums of iid random variables with replacements are obtained and discussed. Also we obtained functional limit theorems for integrals of such random processes. We use our results to study a number of models of the financial market.

1. Introduction

The paper deals with random step lines processes and random broken lines processes defined by sums of independent identically distributed random variables multiplied by values of indicators defined on another probability space. These processes describe some models in which random variables are replaced with other one. We prove our theorems in the Skorokhod space D_{b}(R_{+}) of bounded functions defined on R_{+} = [0, ∞) and in the space C_{b}(R_{+}) of bounded continuous functions defined on R_{+}.

For the indicators we will assume that one of the following properties is valid:

1) in each line the indicators are independent;

2) the indicators are defined by occupations of series of balls.

And our limit theorems are founded on strong laws of large numbers for the indicators (Lemma 1, Lemma 2 and there Corollaries). Observe, that in the case 2) weak law of large numbers for the indicators follows from Theorem 1 on p. 22 in (Kolchin, Sevast’ianov, and Chistiakov, [1]). So Lemma 2 is a straightness of this result.

We have different types of replacements:

a) summands are replaced randomly by zeros;

b) summand are replaced randomly by nonzero random variables.

Also we consider various examples which show that the class of such indicators is sufficiently large. In the case of the convergence to gaussian process we prove the convergence for integrals of our processes in the space C[0, T], 0 < T < ∞, of bounded continuous functions defined on [0, T].

As a corollaries we obtain functional limit theorems in the case then indicators and random variables are defined on the same probability space and are independent. In this case our limit theorems can be considered as functional limit theorems for random sums (see about limit theorems for random sums, for example Korolev and Kruglov [2]).

We give some applications of our results to some models of stochastic financial mathematics. We construct three models of a financial market: the market with constant number of agents, the market with increasing number of agents and the market with decreasing number of agents. We suppose that the trading policy is changing during the time and we consider various types of this changing. For such models we construct the random processes of stock market price with respect to market probability and we construct the random processes of stock market price with respect to risk-neutral probability.

The limit theorems for such models for almost all sequences of the trade policy changing are given. Using this limit theorems we obtained analogs of Black-Scholes formula for there models.

For the space C_{b}(R_{+}) the method of the proofs of our theorems is the same as those of Prokhorov theorem (see Prokhorov, [3]) and for the space D_{b}(R_{+}) the method of the proofs of our theorems is the same as those of Skorokhod theorem (see Gihman and Skorokhod, [4]). Our results generalizes functional limit theorems from Fazekas and Chuprunov [5]. We mention that in Rusakov [6], another type of replacement was considered and a functional limit theorem with an Ornstein-Uhlenbeck limit process was proved.

2. Preliminary results: strong laws of large numbers.

We will use the following denotation: ⟶^{d} – the convergence in distribution; ⟶^{P} – the convergence in probability; =^{d} – the equality in distribution (for random processes =^{d} is the equality of finite dimensional distributions); IN – the set of positive integers; Q – the set of rational numbers; Q_{+} = IR_{+} ∩ Q; Q_{T} – the set of numbers yT where y ∈ [0, 1] ∩ Q;

[c] – integer part of a real number c; {c} – fractional part of a real number c; L(X) – the law of distribution of a random element or a random processes X; 1I_{A} – the indicator function of an event A; γ(v) – a gaussian random variable with the expectation zero and the variance v^{2}; Φ – distribution function of γ(1); σ^{2}(ξ) – the variance of the random variable ξ; W (t), W′(t) – the independent standard Brownian motions (Bm); W_{D}(t), W_{D′}(t), t ∈ IR_{+} – the independent homogeneous infinitely divisible random processes with independent increments such that W_{D}(1) =^{d} γ_{D} and W_{D′}(1) =^{d} γ_{D′} where γ_{D} and γ_{D′} is an infinitely divisible random variables.

For any (a_{i}) we will suppose that ∑_{i∈∅} a_{i} = 0 and ∏_{i∈∅} a_{i} = 1.

We will consider two probability spaces {Ω, 𝔄, P} and {Ω_{1}, 𝔄_{1}, P_{1}}. The first probability space will play a role of a space of noise, and the second one will play a role a space of information. Next we denote by E and E_{1} the mathematical expectations respectively to the probabilities P and P_{1}.

We will consider a scheme of series parameterized by a natural n. Throughout the paper we will use a previously fixed natural parameter m playing a role of viscosity for an information. It shows a rate of changing of informational events along the discrete time. We will examine an asymptotic under a sequence (k_{n}) of natural numbers such that k_{n} → ∞ as n → ∞ and k_{n} < k_{n+1}, n ∈ N.

We will denote informational events from 𝔄_{1} as A_{ij} ≡ A_{ij}(n) for i, j ∈ IN, and corresponding indicators will denote as 1I_{A_{ij}(n)}(ω_{1}) =^{△} 1I_{ij} ≡ 1I_{ij}(n), ω_{1} ∈ Ω_{1}. We will suppose that probabilities of these events independent of the time index i, i.e. the sequence (informational flow) (A_{ij})_{i=1}^{∞} is homogeneous on the time,

P_{1}{A_{ij}(n)} =^{△} p_{j}(n) ≡ p_{j}, ∀i ∈ N

In connect with some ways an interpretation and a formalization an information flow we will suppose that the informational events A_{ij}(n) satisfy to one of the following properties:

(A) A_{ij}(n) are mutually independent over all n ∈ IN and such that for all x ∈ IR_{+} for a continues function f the following limit relation holds and the corresponding limits exist

f(x) =^{△} lim_{n→∞} E_{1}{1I_{i1}(n)1I_{i2}(n) ⋯ 1I_{i[k_{n}x]}(n)} = lim_{n→∞} p_{1}(n)p_{2}(n) ⋯ p_{[k_{n}x]}(n)

In particular, (A) implies that f(0) = 1 and f is decreasing.

(A1) Let ξ, ξ_{j}, j ∈ N be independent identically uniformly distributed on [0, 1) random variables defined on (Ω_{1}, 𝔄_{1}, P_{1}), Δ_{i} = Δ_{ni} = [\frac{i−1}{k_{n}}, \frac{i}{k_{n}}), 1 ≤ i ≤ k_{n} and m ∈ N.

Define the informational events with connection to the split Δ of the interval [0, 1)

A_{ij} ≡ A_{ij}(n, m) = ∩_{k=m(j−1)+1}^{mj} {ω_{1} ∈ Ω_{1} : ξ_{k}(ω_{1}) ∉ Δ_{i}} , 1 ≤ i ≤ k_{n}, j, n ∈ N.

For defined here information events there exists a natural interpretation. The intervals Δ_{j}, j = 1, …, k_{n} we treat as a row of boxes. Random variables ξ_{j} are independent copies of rv ξ. Each realization of rv ξ we treat as random allocation one ball onto the k_{n} boxes. We consider series containing m balls. And the equality 1I_{ij} to one means that for a random occupation all balls from jth series do not allocate the ith box. We will use the following simple corollaries of well-known results.

Lemma 1. Let g_{ni} : Ω_{1} → R, 1 ≤ i ≤ k_{n}, n ∈ N be independent identically distributed random variables in each series such that for all 1 ≤ i ≤ k_{n}, n ∈ N, E_{1}(g_{ni} − E_{1}g_{ni})^{4} ≤ C for some C > 0 and E_{1}g_{in} → p as n → ∞. Then we have

\frac{1}{k_{n}} ∑_{i=1}^{k_{n}} g_{ni} → p as n → ∞

almost surely.

Proof. Observe that E_{1}|\frac{1}{k_{n}} ∑_{i=1}^{k_{n}}(g_{ni} − E_{1}g_{ni})|^{4} ≤ \frac{7C}{k_{n}^{2}} , n ∈ N. Therefore

for all ε > 0 it holds ∑_{n=1}^{∞} P_{1}\{|\frac{1}{k_{n}} ∑_{i=1}^{k_{n}}(g_{ni} − E_{1}g_{ni})| > ε\} ≤ \frac{1}{ε^{4}} ∑_{n=1}^{∞} \frac{7C}{k_{n}^{2}} < ∞.

Consequently, \frac{1}{k_{n}} ∑_{i=1}^{k_{n}}(g_{ni} − E_{1}g_{ni}) → 0 as n → ∞ almost sure. Then we have

\frac{1}{k_{n}} ∑_{i=1}^{k_{n}} g_{ni} − p = \frac{1}{k_{n}} ∑_{i=1}^{k_{n}}(g_{ni} − E_{1}g_{ni}) + E_{1}g_{ni} − p → 0

as n → ∞ almost sure. The proof is complete.

We will set

f_{1n}(t) = f_{1n}(t)(ω_{1}) = ∑_{i=1}^{k_{n}} 1I_{[k_{n}t]i}(n)1I_{([k_{n}x]−1)i}(n) ⋯ 1I_{1i}(n), t ∈ R_{+}, ω_{1} ∈ Ω_{1}

and

f_{2n}(t) = f_{2n}(t)(ω_{1}) = ∑_{i=1}^{k_{n}}(1−1I_{[k_{n}t]i}(n)1I_{([k_{n}x]−1)i}(n) ⋯ 1I_{1i}(n)), t ∈ R_{+}, ω_{1} ∈ Ω_{1},

where n ∈ N.

Corollary 1. Let (A) be valid. Then there exists Ω′ ⊂ Ω_{1} such that P_{1}(Ω′) = 1 and for all ω_{1} ∈ Ω′ uniformly by t ∈ R_{+} one has

\frac{f_{1n}(t)(ω_{1})}{k_{n}} → f(t) as n → ∞.  (1)

Proof. By Theorem A there exists Ω′ ⊂ Ω_{1} such that P_{1}(Ω′) = 1 and for all ω_{1} ∈ Ω′ for all t ∈ Q_{+} (1) is valid.

Let ω_{1} ∈ Ω′ and t ∈ R_{+}. Choose t_{1}, t_{2} ∈ Q_{+} such that 0 ≤ t_{1} ≤ t ≤ t_{2}. Since f_{1n} is a decreasing function we have

\frac{f_{1n}(t_{2})(ω_{1})}{k_{n}} ≤ \frac{f_{1n}(t)(ω_{1})}{k_{n}} ≤ \frac{f_{1n}(t_{1})(ω_{1})}{k_{n}}.

Therefore, one has

f^{∗}(t_{2}) ≤ lim inf_{n→∞} \frac{f_{1n}(t)(ω_{1})}{k_{n}} ≤ lim sup_{n→∞} \frac{f_{1n}(t)(ω_{1})}{k_{n}} ≤ f(t_{1}).

By the continuity of f(t) this follows (1) uniformly by t ∈ R_{+}. The corollary is proved.

Corollary 2. Let (A) be valid. There exists Ω′ ⊂ Ω_{1} such that P_{1}(Ω′) = 1 and for all ω_{1} ∈ Ω′ uniformly by t ∈ R_{+} one has

\frac{f_{2n}(t)(ω_{1})}{k_{n}} → 1 − f(t) as n → ∞.

Since f_{2n}(t) = k_{n} − f_{1n}(t), t ∈ R_{+}, Corollary 2 follows from Corollary 1. From Theorem 1 on p. 22 in (Kolchin, Sevast’ianov, and Chistiakov, [1]) it

follows that for all t > 0 \frac{1}{k_{n}} ∑_{i=1}^{k_{n}} ∏_{j=1}^{[tk_{n}]} I_{pji} → e^{−mt}, as n → ∞ in probability. Consider more strong version of this result.

Lemma 2. Let (A1) be valid. For all t ∈ R_{+} we have

\frac{1}{k_{n}} ∑_{i=1}^{k_{n}} ∏_{j=1}^{[tk_{n}]} 1I_{pji} → e^{−mt}, as n → ∞.

almost sure.

Proof. Let t ∈ R_{+}. Denote g_{ni} = ∏_{j=1}^{[tk_{n}]} 1I_{pji},

g_{i} = g_{ni} − E_{1}g_{ni} = g_{ni} − (1 − \frac{1}{k_{n}} )^{m[tk_{n}]} .

We will estimate

I = E_{1}(∑_{i=1}^{k_{n}} g_{i})^{4} = ∑_{j_{1}=1}^{k_{n}} ∑_{j_{2}=1}^{k_{n}} ∑_{j_{3}=1}^{k_{n}} ∑_{j_{4}=1}^{k_{n}} Eg_{j_{1}}g_{j_{2}}g_{j_{3}}g_{j_{4}}

= k_{n}E(g_{1})^{4} + 6k_{n}(k_{n} − 1)E(g_{1})^{2}(g_{2})^{2} + 4k_{n}(k_{n} − 1)E(g_{1})^{3}g_{2}

+ 12k_{n}(k_{n} − 1)(k_{n} − 2)E(g_{1})^{2}g_{2}g_{3} + 24k_{n}(k_{n} − 1)(k_{n} − 2)(k_{n} − 3)Eg_{1}g_{2}g_{3}g_{4}

= I_{1} + 6I_{2} + 4I_{3} + 12I_{4} + 24I_{5}.

Since |g_{i}| ≤ 1, we have

I_{1} + 6I_{2} + 4I_{3} ≤ k_{n} + 6k_{n}(k_{n} − 1) + 4k_{n}(k_{n} − 1) ≤ 10k_{n}^{2}.  (2)

Using the inequality a^{k} − b^{k} ≤ k(a − b), 0 ≤ b ≤ a ≤ 1, k ∈ N and the inequality

(1 − \frac{1}{k_{n}} )^{2m[tk_{n}]}(1 − \frac{2}{k_{n}} )^{m[tk_{n}]} − (1 − \frac{1}{k_{n}} )^{4[tk_{n}]} ≤ 0

we obtain

I_{4} ≤ k_{n}^{3}E(∏_{i=1}^{[tk_{n}]} 1I_{\{ξ_{i}∉Δ_{1}\}} − (1 − \frac{1}{k_{n}} )^{m[tk_{n}]})^{2}

× (∏_{i=1}^{[tk_{n}]} 1I_{\{ξ_{i}∉Δ_{2}\}} − (1 − \frac{1}{k_{n}} )^{m[tk_{n}]})(∏_{i=1}^{[tk_{n}]} 1I_{\{ξ_{i}∉Δ_{3}\}} − (1 − \frac{1}{k_{n}} )^{m[tk_{n}]})

= k_{n}^{3}E(∏_{i=1}^{[tk_{n}]} 1I_{\{ξ_{i}∉Δ_{1}\}} − (2 ∏_{i=1}^{[tk_{n}]} 1I_{\{ξ_{i}∉Δ_{1}\}})(1 − \frac{1}{k_{n}} )^{m[tk_{n}]} + (1 − \frac{1}{k_{n}} )^{2m[tk_{n}]})

× (∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{2}∪Δ_{3}\}} − (∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{2}\}})(1 − \frac{1}{k_{n}} )^{m[tk_{n}]}

− (∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{1}\}})(1 − \frac{1}{k_{n}} )^{m[tk_{n}]} + (1 − \frac{1}{k_{n}} )^{2m[tk_{n}]})

≤ k_{n}^{3}E((∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{1}\}})(1 − 2(1 − \frac{1}{k_{n}} )^{m[tk_{n}]}) + (1 − \frac{1}{k_{n}} )^{2m[tk_{n}]})

× (∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{2}∪Δ_{3}\}} − (∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{2}\}})(1 − \frac{1}{k_{n}} )^{m[tk_{n}]}

− ∏_{i=1}^{[tk_{n}]} I_{\{ξ_{i}∉Δ_{3}\}}(1 − \frac{1}{k_{n}} )^{m[tk_{n}]} + (1 − \frac{1}{k_{n}} )^{2m[tk_{n}]})

= k_{n}^{3}((1 − 2(1 − \frac{1}{k_{n}} )^{m[tk_{n}]})((1 − \frac{3}{k_{n}} )^{m[tk_{n}]}

− 2(1 − \frac{2}{k_{n}} )^{m[tk_{n}]}(1 − \frac{1}{k_{n}} )^{m[tk_{n}]} + (1 − \frac{1}{k_{n}} )^{3m[tk_{n}]})

+ (1 − \frac{1}{k_{n}} )^{2m[tk_{n}]}(1 − \frac{2}{k_{n}} )^{m[tk_{n}]} − (1 − \frac{1}{k_{n}} )^{4m[tk_{n}]})

≤ n^{3}((1 − \frac{1}{k_{n}} )^{3m[tk_{n}]} − (1 − \frac{2}{k_{n}} )^{m[tk_{n}]}(1 − \frac{1}{k_{n}} )^{m[tk_{n}]})
≤ k_{n}^{3}((1 − \frac{2}{k_{n}} + \frac{1}{k_{n}^{2}} )^{m[tk_{n}]} − (1 − \frac{2}{k_{n}} )^{m[tk_{n}]}) ≤ k_{n}^{3} \frac{1}{k_{n}^{2}} m[tk_{n}] ≤ mtk_{n}^{2}. (3)

CONVERGENCE FOR STEP LINE PROCESSES 17

Now, using the Newton binomial, we have

I_{5} ≤ k_{n}^{4}Eg_{1}g_{2}g_{1}g_{3}g_{4} = k_{n}^{4}((1 − \frac{4}{k_{n}} )^{m[tk_{n}]} − 4(1 − \frac{3}{k_{n}} )^{m[tk_{n}]}(1 − \frac{1}{k_{n}} )^{m[tk_{n}]}

+ 6(1 − \frac{2}{k_{n}} )^{m[tk_{n}]}(1 − \frac{1}{k_{n}} )^{2m[tk_{n}]} − 4(1 − \frac{1}{k_{n}} )^{4m[tk_{n}]}

+ (1 − \frac{1}{k_{n}} )^{4m[tk_{n}]})

= k_{n}^{4}((1 − \frac{4}{k_{n}} )^{m[tk_{n}]} − 4(1 − \frac{4}{k_{n}} + \frac{3}{k_{n}^{2}} )^{m[tk_{n}]}

+ 6(1 − \frac{4}{k_{n}} + \frac{5}{k_{n}^{2}} − \frac{2}{k_{n}^{3}} )^{m[tk_{n}]} − 3(1 − \frac{4}{k_{n}} + \frac{6}{k_{n}^{2}} − \frac{4}{k_{n}^{3}} + \frac{1}{k_{n}^{4}} )^{m[tk_{n}]})

= k_{n}^{4}((1 − \frac{4}{k_{n}} )^{m[tk_{n}]} − 4 ∑_{k=0}^{m[tk_{n}]} C_{m[tk_{n}]}^{k}(1 − \frac{4}{k_{n}} )^{m[tk_{n}]−k}(\frac{3}{k_{n}^{2}} )^{k}

+ 6 ∑_{k=0}^{m[tk_{n}]} C_{m[tk_{n}]}^{k}(1 − \frac{4}{k_{n}} )^{m[tk_{n}]−k}(\frac{5}{k_{n}^{2}} − \frac{2}{k_{n}^{3}} )^{k}

− 3 ∑_{k=0}^{m[tk_{n}]} C_{m[tk_{n}]}^{k}(1 − \frac{1}{k_{n}} )^{m[tk_{n}]−k}(\frac{6}{k_{n}^{2}} − \frac{4}{k_{n}^{3}} + \frac{1}{k_{n}^{4}} )^{k})

= k_{n}^{4}(k_{n}(−\frac{12}{k_{n}^{2}} + \frac{30}{k_{n}^{2}} − \frac{12}{k_{n}^{3}} − \frac{18}{k_{n}^{2}} + \frac{12}{k_{n}^{3}} − \frac{3}{k_{n}^{4}} )(1 − \frac{1}{k_{n}} )^{m[tk_{n}]−1}

+ ∑_{k=2}^{m[tk_{n}]} C_{m[tk_{n}]}^{k}[−4(\frac{3}{k_{n}^{2}} )^{k}+6(\frac{5}{k_{n}^{2}} − \frac{2}{k_{n}^{3}} )^{k}−3(\frac{6}{k_{n}^{2}} − \frac{4}{k_{n}^{3}} + \frac{1}{k_{n}^{4}} )^{k}]).

Since

k_{n}(−\frac{12}{k_{n}^{2}} + \frac{30}{k_{n}^{2}} − \frac{12}{k_{n}^{3}} − \frac{18}{k_{n}^{2}} + \frac{12}{k_{n}^{3}} − \frac{3}{k_{n}^{4}} )(1 − \frac{1}{k_{n}} )^{m[tk_{n}]−1} < 0

and

−4(\frac{3}{k_{n}^{2}} )^{k}+6(\frac{5}{k_{n}^{2}} − \frac{2}{k_{n}^{3}} )^{k}−3(\frac{6}{k_{n}^{2}} − \frac{4}{k_{n}^{3}} + \frac{1}{k_{n}^{4}} )^{k}< 0

for k > 4, we have

I_{5} ≤ k_{n}^{4} ∑_{k=2}^{4} C_{m[tk_{n}]}^{k}[−4(\frac{3}{k_{n}^{2}} )^{k}+6(\frac{5}{k_{n}^{2}} − \frac{2}{k_{n}^{3}} )^{k}−3(\frac{6}{k_{n}^{2}} − \frac{4}{k_{n}^{3}} + \frac{1}{k_{n}^{4}} )^{k}] ≤

≤ \frac{k_{n}^{4}}{k_{n}^{2}} ∑_{k=2}^{4} \frac{C_{m[tk_{n}]}^{k}}{k_{n}^{2(k−1)}} 6 · 5^{k} ≤ 6(t^{2}m^{2} \frac{25}{2} + t^{3}m^{3} \frac{125}{6k_{n}} + t^{4}m^{4} \frac{625}{24k_{n}^{2}} )k_{n}^{2}. (4)

From (2), (3) and (4) we obtain

E(∑_{i=1}^{k_{n}}(g_{ni} − E_{1}g_{ni}))^{4} ≤ C′k_{n}^{2},

where C′ does not depends from n. Further the proof repeats the proof of Lemma 1.

Corollary 1. Let (A1) be valid. Then there exists Ω′ ⊂ Ω_{1} such that P_{1}(Ω′) = 1 and for all ω_{1} ∈ Ω′ uniformly for t ∈ R_{+} one has

\frac{f_{1n}(t)(ω_{1})}{k_{n}} → e^{−mx} as n → ∞.

Corollary 2. Let (A1) be valid. Then there exists Ω′ ⊂ Ω_{1} such that P_{1}(Ω′) = 1 and for all ω_{1} ∈ Ω′ uniformly for x ∈ R^{+} one has

\frac{f_{2n}(t)(ω_{1})}{k_{n}} → 1 − e^{−mt} as n → ∞.

The proofs of Corollary 1 and Corollary 2 is the same as ones of Lemma 1. We will denote f_{m}(t) = e^{−mt}, t ∈ R_{+}.

3. Preliminary results: functional limit theorems To construe our noise structure and to study the random processes we give

conditions and definitions for considered random elements on {Ω, 𝔄, P}.

(Y) Let Y_{n}, Y_{ni}, n, i ∈ N be an rectangle array of random variables defined on {Ω, 𝔄, P} which are independent and identically distributed in each string.

(Y’) Let Y′_{n}, Y′_{ni}, i ∈ N be an array of independent and identically distributed in each string random variables defined on (Ω, 𝔄, P) such that Y′_{ni}, Y_{ni}, i ∈ N are independent random variables for all n ∈ N.

And we will assume that some of the following conditions are satisfied:

(C) ∑_{j=1}^{k_{n}} Y_{nj} ⟶^{d} γ_{D}, as n → ∞;  (C’) ∑_{j=1}^{k_{n}} Y′_{nj} ⟶^{d} γ_{D′}, as n → ∞;

(S) ∑_{i=1}^{k_{n}} Y_{ni} ⟶^{d} γ(v), as n → ∞; (S’) ∑_{i=1}^{k_{n}} Y′_{ni} ⟶^{d} γ(v′), as n → ∞.

The following theorem is a simple corollary of Lindeberg-Feller Theorem. Theorem A. Let (Y) be fulfilled.

(1) The condition (S) is valid if and only if

(1b) for all ε > 0 k_{n}P{|Y_{n}| > ε} → 0 as n → ∞;

(2b) k_{n}EY_{n}1I_{{|Y_{n}|≤1}} → 0 as n → ∞; and (3b) k_{n}σ^{2}(Y_{n}1I_{{|V_{n}|≤1}}) → v^{2}, as n → ∞.

(2) Let (S) be valid and b_{ni} ∈ R be such that sup_{{1≤i≤k_{n},n∈N}} |b_{ni}| < ∞.

Denote U_{n} = ∑_{i=1}^{k_{n}} b_{ni}Y_{ni}, n ∈ N. Then U_{n} ⟶^{d} γ(s) as n → ∞ if and only if

(4b) σ^{2}(Y_{n}1I_{{|Y_{n}|≤1}}) ∑_{i=1}^{k_{n}}(b_{ni})^{2} → s^{2}, as n → ∞.

(D) Let f_{n}(x), f^{∗}(x) : IR_{+} ↦ IR_{+}, n ∈ IN be integer valued functions, f^{∗}(x) is a continue bounded function plays a role of the following limit

\frac{f_{n}(x)}{k_{n}} → f^{∗}(x), as n → ∞

for all x ∈ [0, ∞] and one of the following conditions is true:

a) f_{n} are increasing functions;

b) f_{n} are decreasing functions.

In the paper we will examine a sequence of the random processes with a time scale managing by the functions (f_{n}(x)),

(Z0′)  X′_{∗n}(x) = X′_{∗n}(Y)(x) = ∑_{i=1}^{f_{n}(x)} Y_{ni}, x ∈ IR_{+},  n ∈ IN

and their modifications formed by piece-wise linear continuous broken lines,

(Z0) X_{0n}(x) = X_{0n}(Y)(x) = X′_{∗n}(x_{k}) + \frac{x − x_{k}}{x_{k+1} − x_{k}} (X′_{∗n}(x_{k+1}) − X′_{∗n}(x_{k})) ,

n ∈ IN, x ∈ [x_{k}, x_{k+1}), where 0 = x_{0} < x_{1} < x_{2} … are the points of jumps of f_{n}.

Theorem B. Let the conditions (C) and (D) be valid. Then for the processes, defined by (Z0’) one has

X′_{∗n} →^{d} W_{f^{∗}D}, as n → ∞

in D_{b}(R_{+}) where W_{f^{∗}D}(x) = W_{D}(f^{∗}(x)), x ∈ R_{+}.

The proof of Theorem B is the same as one of Skorokhod Theorem (see Theorem 1 on p. 547, [4]).

Theorem C. Let the conditions (S) and (D) be valid. Then for the processes, defined by (Z0) one has

X_{0n} →^{d} W_{f^{∗}v}, as n → ∞

in C_{b}(R_{+}), where W_{f^{∗}v}(x) = W (v^{2}f^{∗}(x)), x ∈ R_{+}.

Proof. We represent X_{0n} as follows

X_{0n} = X_{0n}(Y^{(1)}) + X_{0n}(Y^{(2)}) + X_{0n}(Y^{(3)}),

where Y_{n}^{(3)} = Y_{n}I_{{|Y_{n}|>1}}, Y_{n}^{(2)} = EY_{n}1I_{{|Y_{n}|≤1}}, and Y_{n}^{(1)} = Y_{n}1I_{{|Y_{n}|≤1}} − EY_{n}1I_{{|Y_{n}|≤1}}. Since for all ε > 0 it holds

P{ sup_{t∈R_{+}} |X_{0n}(Y^{(3)})| > ε} < k_{n}P{|Y_{n}| > 1},

by (1b) we have X_{0n}(Y^{(3)}) → 0, as n → ∞ in probability in C_{b}(R_{+}). If ε > k_{n}|EY_{n}1I_{{|Y_{n}|≤1}}|, then

P{ sup_{t∈R_{+}} |X_{0n}(Y^{(2)})| > ε} = 0.

Therefore by (2b) one has X_{0n}(Y^{(2)}) → 0, as n → ∞ in probability in C_{b}(R_{+}). The random variables Y_{ni}^{(1)} satisfy to the conditions of Lindeberg-Feller theorem. Consequently, repeating the proof of Prokhorov Theorem [3] we can show that X_{n}^{(1)} →^{d} W_{fv} as n → ∞ in C_{b}(R_{+}). This implies Theorem D.

It is easy to see that the correlation function of W_{f^{∗}v} in the case of a) is B_{0}(x, y) = v^{2}f^{#}(x, y), where f^{#}(x, y) = f^{∗}(min(x, y)) and the correlation function of W_{f^{∗}v} in the case of b) is B_{0}(x, y) = v^{2}f^{#}(x, y), where f^{#}(x, y) = f^{∗}(max(x, y))

Formulate the analogy of (D):

(D’) Let f_{n}^{∗∗}(x), f^{∗∗}(x) : IR_{+} ↦ IR_{+}, n ∈ IN be integer valued functions, f^{∗∗}(x) is a continue bounded function plays a role of the following limit

\frac{f_{n}^{∗∗}(x)}{k_{n}} → f^{∗∗}(x), as n → ∞

for all x ∈ [0, ∞] x ∈ IR_{+} and one of the following conditions is true: a) f_{n}^{∗∗} are increasing functions;

b) f_{n}^{∗∗} are decreasing functions.

We will consider the random processes

(Z0”)  X′_{∗∗n}(x) = X′_{∗∗n}(Y′)(x) = ∑_{i=1}^{f_{n}^{∗∗}(x)} Y′_{ni} ,  n ∈ IN  x ∈ IR_{+}.

and their modifications formed by piece-wise linear continuous broken lines, (Z0^{∗})

X_{∗∗n}(x) = X_{∗∗n}(Y′)(x) = X′_{∗∗n}(x_{k}) + \frac{x − x_{k}^{∗}}{x_{k+1}^{∗} − x_{k}^{∗}} (X′_{∗∗n}(x_{k+1}^{∗}) − X′_{∗∗n}(x_{k}^{∗})) ,

n ∈ IN, x ∈ [x_{k}^{∗}, x_{k+1}^{∗}), where 0 = x_{0}^{∗} < x_{1}^{∗} < x_{2}^{∗} … are the points of jumps of f_{n}.

Since X_{0n} and X_{∗∗n} are independent random processes from Theorem C it follows

Theorem D. Let the conditions (S), (S’), (D) and (D’) be valid. Then for the processes, defined by (Z0) and (Z0^{∗}) one has

X_{0n} + X_{∗∗n} →^{d} W_{f^{∗}v} + W_{f^{∗∗}v′}, as n → ∞

in C_{b}(R_{+}).

Let −∞ ≤ a < b ≤ ∞, For a function x : [a, b] → R_{+} we define

Δ_{a,b}(x) = sup_{a≤t′≤t≤t”≤b}[min{|x(t′) − x(t)|, |x(t) − x(t”)|}].

In order to prove a version of Theorem D for Skorokhod space we need the following lemma.

Lemma 3.Let V′(t) and V ”(t), t ∈ [a, b] be separable independent random processes. Denote V = V′ + V ”. Then for all ε > 0 we have

P {Δ_{a,b}(V) > ε}

≤ 2P \{ sup_{a≤t≤b} |V′(b) − V′(t)| > \frac{ε}{4}\} P \{ sup_{a≤t≤b} |V ”(b) − V ”(t)| > \frac{ε}{4}\}

+ P \{Δ_{a,b}(V′) > \frac{ε}{2}\} + P \{Δ_{a,b}(V ”) > \frac{ε}{2}\} .

Proof. Using the inequality |a + b| ≤ 2 max(|a|, |b|), a, b ∈ R_{+}, we obtain

P {Δ_{a,b}(V) > ε} ≤ P{2 sup_{a≤t′≤t≤t”≤b}[min{max(|V′(t′) − V′(t)|, |V ”(t′) − V ”(t)|),

max(|V′(t) − V′(t”)|, |V ”(t) − V ”(t”)|)}] > ε}

≤ P \{Δ_{a,b}(V′) > \frac{ε}{2}\} + P \{Δ_{a,b}(V ”) > \frac{ε}{2}\}

× P\{ sup_{a≤t′≤t≤t”≤b}[min\{|V′(t′) − V′(t)|, |V ”(t) − V ”(t”)|\}] > \frac{ε}{2}\}

× P\{ sup_{a≤t′≤t≤t”≤b}[min\{|V ”(t′) − V ”(t)|, |V′(t) − V′(t”)|\}] > \frac{ε}{2}\}

≤ 2P \{ sup_{a≤t≤b} |V′(b) − V′(t)| > \frac{ε}{4}\} P \{ sup_{a≤t≤b} |V ”(b) − V ”(t)| > \frac{ε}{4}\}

+ P \{Δ_{a,b}(V′) > \frac{ε}{2}\} + P \{Δ_{a,b}(V ”) > \frac{ε}{2}\} .

The proof is complete.

The addition in Skorokhod space is not continuous. Therefore in general the sum of convergence sequences of random processes does not appear a convergence sequence. However we have

Theorem E. Let the conditions (C), (C’), (D) and (D’) be valid. Then for the processes, defined by (Z0’) and (Z0”) one has

X′_{∗n} + X′_{∗∗n} →^{d} W_{f^{∗}D} + W_{f^{∗∗}D′}, as n → ∞

in D_{b}(R_{+}).

Proof. Since by Theorem C X′_{∗n} →^{d} W_{f^{∗}D}, as n → ∞ and X′_{∗∗n} →^{d} W_{f^{∗∗}D′}, as n → ∞ and for all n ∈ N X′_{∗n} and X′_{∗∗n} are independent in D_{b}(R_{+}) and for all n ∈ N X′_{∗n} and X′_{∗∗n} are independent, finite dimensional distributions of X′_{∗n} + X′_{∗∗n} converge to finite dimensional distributions of W_{f^{∗}D} + W_{f^{∗∗}D′}.

The proof of the tightness of {L(X′_{∗n} + X′_{∗∗n}), n ∈ N} in D_{b}(R_{+}) is the same as Skorokhod Theorem (see Theorem 1 on p. 548, [3]). Only with the Skorokhod inequality (see Lemma 1 on p. 548, [4]) we use Lemma 3. By Theorem 2 (see p. 545, [4]) this implies Theorem E.

Now we will prove a functional limit theorem for integral of X′_{0n}. More precisely we will consider the random processes:

(Z0)

Z_{0n}(t) ≡ Z_{0n}(t)(Y) = ∫_{0}^{t} X′_{∗n}(x)dx = ∑_{k=1}^{k_{t}} ( ∑_{i=1}^{f_{n}(x_{k−1})} Y_{ni}) Δ_{k} +( ∑_{i=1}^{f_{n}(x_{k_{t}})} Y_{ni}) Δ_{t}

t ∈ IR_{+}, n ∈ IN and the random processes

(Z0’) Z′_{0n}(x) ≡ Z′_{0n}(x)(Y) = ∑_{k=1}^{k_{t}} ( ∑_{i=1}^{f_{n}(x_{k−1})} Y_{ni}) Δ_{k},  t ∈ IR_{+} , n ∈ IN.

Here Δ_{k} = x_{k} − x_{k−1}, k ∈ N, k_{t} = sup{k : x_{k} ≤ t}, x_{t} = x_{k_{t}}, and Δ_{t} = t − x_{t}. We will use the condition:

(E)  max_{1≤k≤k_{T} +1} Δ_{k} → 0 as n → ∞.

Consider the function:

c) f_{n}^{#}(x, y) = \frac{f_{n}(min(x,y))}{k_{n}} in the case of a);

d) f_{n}^{#}(x, y) = \frac{f_{n}(max(x,y))}{k_{n}} in the case of b);

e) g_{n}(x, y) = ∑_{k_{1}=1}^{k_{x}} ∑_{k_{2}=1}^{k_{y}} f_{n}^{#}(x_{k_{1}−1}, x_{k_{2}−1})Δ_{k_{1}}Δ_{k_{2}} and

f) g(x, y) = ∫_{0}^{x} ∫_{0}^{y} f^{#}(t_{1}, t_{2})dt_{1}dt_{2}, x, y ∈ R_{+}.

Lemma 4. Let (D) and (E) be valid. Then for all x, y ∈ R_{+} one has

g_{n}(x, y) → g(x, y), as n → ∞,

Proof. Since f_{n}, f^{∗} are monotone bounded functions, f_{n}(x) → f^{∗}(x), as n → ∞ uniformly by x ∈ R_{+}. Therefore

g_{n}(x, y) − ∑_{k_{1}=1}^{k_{x}} ∑_{k_{2}=1}^{k_{y}} f^{#}(x_{k_{1}−1}, x_{k_{2}−1})Δ_{k_{1}}Δ_{k_{2}} → 0, as n → ∞.

Observe that

∑_{k_{1}=1}^{k_{x}} ∑_{k_{2}=1}^{k_{y}} f^{#}(x_{k_{1}−1}, x_{k_{2}−1})Δ_{k_{1}}Δ_{k_{2}}

is an integral sum for the integral ∫_{0}^{t_{x}} ∫_{0}^{t_{y}} f^{#}(t_{1}, t_{2})dt_{1}dt_{2}. Consequently,

∑_{k_{1}=1}^{k_{x}} ∑_{k_{2}=1}^{k_{y}} f^{#}(x_{k_{1}−1}, x_{k_{2}−1})Δ_{k_{1}}Δ_{k_{2}} → ∫_{0}^{x} ∫_{0}^{y} f^{#}(t_{1}, t_{2})dt_{1}dt_{2}, as n → ∞.

This follows the lemma.

We will use the following version of Levy maximal inequality (see. p. 262,

[7]). Let U_{i}, 1 ≤ i ≤ n, be independent random variables. Denote V_{k} = ∑_{i=1}^{k} U_{i}.

Then for all ε > 0 one has

P\{max_{k≤n} |V_{k}| > ε\} ≤ 2P\{|V_{n}| > ε − \sqrt{2σ^{2}(V_{n})}\}.  (5)

Observe that (5) remain true if instead max_{k≤n} V_{k} the expression max_{k∈M} V_{k}, where M ⊂ {1, 2, … n}. We will use this in the following lemma.

Lemma 5. Let (D), (Y) and (S) be valid. Then for all c, ε > 0 it holds

lim sup_{n→∞} P\{ sup_{0≤t<∞} c|X′_{∗n}(t)| > ε\} ≤ 2P\{c|γ(v_{1})| > \frac{ε}{2} − \sqrt{2}v_{1}c\},

where v_{1} = v sup_{t∈R_{+}} f^{∗}(t).

Proof. Denote m_{n} = sup_{0≤t<∞}f_{n}(x). By (D) \frac{m_{n}}{k_{n}} → sup_{t∈R_{+}} f^{∗}(t) as

n → ∞. Therefore ∑_{i=1}^{m_{n}} Y_{ni} →^{d} γ(v_{1}) as n → ∞. So, by Theorem B(1) using denotation of Theorem D we obtain

lim sup_{n→∞} P{ sup_{0≤t<∞} c|X′_{∗n}(t)| > ε} ≤ lim sup_{n→∞} P{ sup_{1≤k≤m_{n}} c| ∑_{i=1}^{k} Y_{ni}| > ε}

≤ lim sup_{n→∞}(P\{ sup_{1≤k≤m_{n}} c| ∑_{i=1}^{k} Y_{ni}^{(1)}| > \frac{ε}{2}\}+P\{ sup_{1≤k≤m_{n}} c| ∑_{i=1}^{k}(Y_{ni}^{(2)}+Y_{ni}^{(3)})| > \frac{ε}{2}\})

≤ lim sup_{n→∞}(P\{ sup_{1≤k≤m_{n}} c| ∑_{i=1}^{k} Y_{ni}^{(1)})| > \frac{ε}{2}\}

+ P\{c ∑_{i=1}^{m_{n}} |Y_{ni}|1I_{\{|Y_{ni}|>1\}} > \frac{ε}{2} − ck_{n}|EY_{n}1I_{\{|Y_{n}|≤1\}}|\})

≤ 2 lim sup_{n→∞} P\{c| ∑_{i=1}^{k} Y_{ni}^{(1)})| > \frac{ε}{2} − c\sqrt{2σ^{2}(∑_{i=1}^{k} Y_{ni}^{(1)})}\}

= 2P\{c|γ(v_{1})| > \frac{ε}{2} − \sqrt{2}v_{1}c\}.

The lemma is proved. Corollary. Let (D), (E), (Y) and (S) be valid. For all c, ε > 0 and ω_{1} ∈ Ω′

it holds

P{ sup_{t∈[0,T]} |Z_{0n}(t) − Z′_{0n}(t)| > ε} → 0, as n → ∞.

Proof. Observe that

P{ sup_{t∈R_{+}} |Z_{0n}(t) − Z′_{0n}(t)| > ε} ≤ P{2 ( sup_{1≤k≤k_{T} +1} Δ_{k}) ( sup_{0≤k≤m_{n}} | ∑_{i=1}^{m_{n}} Y_{ni}|) > ε}.

By Lemma 5 this implies Corollary. Theorem F. Assume that the conditions (Y), (S), (D) and (E) are fulfilled.

Then the processes defined by (Z0) weakly converge in C[0, T] as follows

Z_{0n} ⟶^{d} W_{0}, as n → ∞,

where W_{0} is a centered gaussian random process with the covariance B_{1}(t_{1}, t_{2}) = v_{1}^{2}g(t_{1}, t_{2}), t_{1}, t_{2} ∈ [0, T].

Proof. Let ε > 0, M ∈ N, 0 < δ < \frac{T}{M} Denote y_{k} = k \frac{T}{M}, 0 ≤ k ≤ M. In the same way as in the proof of Theorem 1 in p. 358 [7], we have

P { sup_{|t′−t”|≤δ} |Z_{0n}(t′) − Z_{0n}(t”)| > ε}

≤ 4 ∑_{k=1}^{M} P \{ sup_{y_{k−1}≤t≤y_{k}} |Z_{0n}(t) − Z_{0n}(y_{k})| > \frac{ε}{8}\}

≤ 4 ∑_{k=1}^{M} P \{ sup_{y_{k−1}≤t≤y_{k}} |∫_{y_{k−1}}^{t} X_{∗n}(t)| > \frac{ε}{8}\}

≤ 4 ∑_{k=1}^{M} P\{\frac{T}{m} sup_{1≤l≤m_{n}} | ∑_{i=i}^{l} Y_{ni}| > \frac{ε}{8}\}.

Thus by Lemma 5 we obtain

lim_{δ→0} lim sup_{n→∞} P { sup_{|t′−t”|≤δ} |Z_{0n}(t′) − Z_{0n}(t”)| > ε}

≤ lim_{M→∞} lim sup_{n→∞} ≤ 4 ∑_{k=1}^{M} P \{\frac{T}{M} sup_{1≤l≤m_{n}} |∑_{i=i}^{l} Y_{ni}| > \frac{ε}{8}\}

≤ 8 lim_{M→∞} MP \{\frac{T}{M} |γ(v_{1})| > \frac{ε}{16} − \sqrt{2}v_{1} \frac{T}{M} \} = 0. (6)

Let M ∈ N, 0 ≤ t_{1} < t_{2} … t_{M} ≤ T and a_{k} ∈ R, 1 ≤ k ≤ M. We will show

∑_{k=1}^{M} a_{k}Z_{0n}(t_{k}) →^{d} ∑_{k=1}^{M} a_{k}W_{0}(t_{k}), as n → ∞.

By Corollary of Lemma 5 it is sufficient to prove that

∑_{k=1}^{M} a_{k}Z′_{0n}(t_{k}) →^{d} ∑_{k=1}^{M} a_{k}W_{0}(t_{k}), as n → ∞.

Consider U_{n} = ∑_{k=1}^{M} a_{k}Z′_{0n}(t_{k}) = ∑_{i=1}^{m_{n}} b_{ni}Y_{ni}. Observe that |b_{ni}| ≤ C, where

C = T ∑_{k=1}^{M} |a_{k}|, and

∑_{i=1}^{m_{n}} E(b_{ni}Y_{ni}^{(1)})^{2} = E (∑_{k=1}^{M} a_{k}Z′_{0n}(Y^{(1)})(t_{k}))^{2} = k_{n}v_{n}^{2} ∑_{l_{1},l_{2}=1}^{M,M} a_{l_{1}}a_{l_{2}}g_{n}(t_{l_{1}}, t_{l_{2}}),

where v_{n}^{2} = σ^{2}(Y_{n}^{(1)}). Therefore by Lemma 3 we have as n → ∞,

∑_{i=1}^{m_{n}} E(b_{ni}Y_{ni}^{(1)})^{2} → v^{2} ( ∑_{l_{1},l_{2}=1}^{M,M} a_{l_{1}}a_{l_{2}}g(t_{l_{1}}, t_{l_{2}})) = E (∑_{k=1}^{M} a_{k}W_{1}(t_{k}))^{2} ,

Consequently, by Theorem B(2) ∑_{k=1}^{M} a_{k}Z_{0n}(t_{k}) →^{d} ∑_{k=1}^{M} a_{k}W_{0}(t_{k}), as n → ∞.

So, by Theorem 7.7 p. 76 in [8] finite dimensional distributions of Z_{0n} converge to finite dimensional distributions of W_{0}. By Theorem 1 on p. 522 in [4] this and (6) imply Z_{0n} →^{d} W_{0} as n → ∞ in C[0, T]. The proof is complete.

Observe that the random processes Z′_{0n} have not continuous trajectories. Therefore it need to consider the convergence of Z′_{0n} in more wide spaces then C[0, T].

Theorem G. Assume that the conditions (D), (E), (Y), and (S) are fulfilled. Then the processes defined by (Z0’) weakly converge in L^{∞}[0, T] as follows

Z′_{0n} ⟶^{d} W_{0}, as n → ∞.

Proof. By Corollary of Lemma 5 Z_{0n}(t) − Z′_{0n}(t) ⟶^{P} 0 as n → ∞ in L^{∞}[0, T]. The space C[0, T] is a subspace of L^{∞}[0, T]. Trajectories of Z′_{0n}(t) belongs to L^{∞}[0, T]. Therefore Theorem G follows from Theorem F. The proof is complete.

4. Main results.

At first we consider the case of the replacement by zeros.

Fix n and describe a recurrent procedure of operations on rows.

(0) We start with initial string (Y_{ni})_{i=1}^{∞}.

(1) At each row i on each term the indicator 1I_{ij} impacts. It means that in the case a term is a rv Y_{ni} remains to ith step non-zero valued (as rv) and 1I_{ij} equals to zero than the term Y_{ni} is disappear at ith step or is replacing with 0, and the term Y_{nj} remains the same if 1I_{ij} = 1.

Define the sequence (S_{k}^{(1n)})_{k=0}^{∞} of row sums,

(X1)  S_{0}^{(1n)} = ∑_{j=1}^{k_{n}} Y_{nj},

S_{1}^{(1n)} = ∑_{j=1}^{k_{n}} 1I_{1j}(n)Y_{nj},

S_{2}^{(1n)} = ∑_{j=1}^{k_{n}} 1I_{2j}(n)1I_{1j}(n)Y_{nj},

⋮

S_{k}^{(1n)} = ∑_{j=1}^{k_{n}} 1I_{1j}(n)1I_{2j}(n) ⋯ 1I_{kj}(n)Y_{nj},

⋮

We will consider sequences of step lines random processes depending from ω_{1} ∈ Ω_{1}. As we shell use the same type of construction in several cases, (Zi′) will denote that the process X′_{in} is defined by the random variables S_{k}^{(in)}, i ∈ N. For i ∈ N let

(Zi’) X′_{in}(x) ≡ X′_{in}(x; ω_{1})(Y) =^{△} S_{[k_{n}x]}^{(in)},  x ∈ IR_{+} , n ∈ IN , ω_{1} ∈ Ω_{1} .

Theorem 1. Assume that the conditions (Y) and (C) are satisfied.

(1) Let (A) be valid. Then the processes defined by (X1) and (Z1’) for almost all ω_{1} ∈ Ω_{1} weakly converge in D_{b}(R_{+}) as follows

X′_{1n}(ω_{1}) ⟶^{d} W_{fD}, as n → ∞.

(2) Let (A2) be valid. Then the processes defined by (X1) and (Z1’) for almost all ω_{1} ∈ Ω_{1} weakly converge in D_{b}(R_{+}) as follows

X′_{1n}(ω_{1}) ⟶^{d} W_{f_{m}D}, as n → ∞.

Observe that X′_{1n} =^{d} X′_{∗n} with f_{n}(x) = f_{1n}(x). So Theorem 1(1) follows from Theorem B and Corollary 1 of Lemma 1, Theorem 1(2) follows from Theorem B and Corollary 1 of Lemma 2.

We will consider the following random broken line sequence:

(Zi)  X_{in}(x) ≡ X_{in}(x; ω_{1})(Y) =^{△} S_{[k_{n}x]}^{(in)} + {k_{n}x}(S_{[k_{n}x]+1}^{(in)} − S_{[k_{n}x]}^{(in)}),

x ∈ IR_{+}, n ∈ IN, ω_{1} ∈ Ω_{1}, i ∈ N.

Since X_{1n} =^{d} X_{0n} with f_{n} = f_{1n}, from Theorem C and Corollary 1 of Lemma 1 or Corollary 1 of Lemma 2, respectively, it follows

Theorem 2. Assume that the conditions (Y) and (S) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (X1) and (Z1) for almost all ω_{1} ∈ Ω_{1} weakly converge in C_{b}(R_{+}) as follows

X_{1n}(ω_{1}) ⟶^{d} W_{fv}, as n → ∞.

(2) Let (A1) be valid. Then the processes defined by (X1) and (Z1) for almost all ω_{1} ∈ Ω_{1} weakly converge in C_{b}(R_{+}) as follows

X_{1n}(ω_{1}) ⟶^{d} W_{f_{m}v}, as n → ∞.

In Theorem 1 and Theorem 2 where considered the case of sums in which the number of summands is decreasing. Now we will consider the case in which the number of summands in sums is increasing. Let

(X2)  S_{0}^{(2n)} = 0,

S_{1}^{(2n)} = ∑_{i=1}^{k_{n}}(1 − 1I_{1i}(n))Y′_{ni},

S_{2}^{(2n)} = ∑_{i=1}^{k_{n}}(1 − 1I_{1i}(n)1I_{2i}(n))Y′_{ni},

⋮

X_{k}^{(2n)} = ∑_{i=1}^{k_{n}}(1 − 1I_{1i}(n)1I_{2i}(n) ⋯ 1I_{ki}(n))Y′_{ni},

⋮

Observe that X′_{2n} =^{d} X′_{∗n} and X_{2n} =^{d} X_{0n} with f_{n}(x) = f_{2n}(x). Therefore by Corollary 2 of Lemma 1 or Corollary 2 of Lemma 2, Theorem D or Theorem E respectively we obtain

Theorem 3. Assume that the conditions (Y) and (C) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (X2) and (Z2’) weakly converge in D_{b}(R_{+}) as follows

X′_{2n}(ω_{1}) ⟶^{d} W_{(1−f)D′}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

(2) Let (A1) be valid. Then the processes defined by (X2) and (Z2’) weakly converge in D_{b}(R_{+}) as follows

X′_{2n}(ω_{1}) ⟶^{d} W_{(1−f_{m})D′}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

Theorem 4. Assume that the conditions (Y) and (S) are valid.

(1) Let (A) be valid. Then the processes defined by (X2) and (Z2) weakly converge in C_{b}(R_{+}) as follows

X_{2n}(ω_{1}) ⟶^{d} W_{(1−f)v′}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

(2) Let (A1) be valid. Then the processes defined by (X2) and (Z2) weakly converge in C_{b}(R_{+}) as follows

X_{2n}(ω_{1}) ⟶^{d} W_{(1−f_{m})v′}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

Now we consider the case of the replacement by non-zero random variables. We start with the summands Y_{ni}, 1 ≤ i ≤ k_{n} and in each step, in the random

way, we change Y_{ni} to Y′_{ni}, 1 ≤ i ≤ k_{n}. More precisely, let

(X3)  S_{0}^{(3n)} = ∑_{i=1}^{k_{n}} Y_{ni},

S_{1}^{(3n)} = ∑_{i=1}^{k_{n}} 1I_{1i}(n)Y_{ni} + ∑_{i=1}^{k_{n}}(1 − 1I_{1i}(n))Y′_{ni},

S_{2}^{(3n)} = ∑_{i=1}^{k_{n}} 1I_{1i}(n)1I_{2i}(n)Y_{ni} + ∑_{i=1}^{k_{n}}(1 − 1I_{1i}(n)1I_{2i}(n))Y′_{ni},

⋮

S_{k}^{(3n)} = ∑_{i=1}^{k_{n}} 1I_{1i}(n)1I_{2i}(n) ⋯ 1I_{ki}(n)Y_{ni} + ∑_{i=1}^{k_{n}}(1 − 1I_{1i}(n)1I_{2i}(n) ⋯ 1I_{ki}(n))Y′_{ni},

⋮

Since X_{3n} = X_{1n}(Y) + X_{2n}(Y′) and X_{1n}(Y), X_{2n}(Y′) are independent random processes, from Theorem D and Theorem E, respectively, it follows

Theorem 5. Suppose that (Y), (S), (Y’) and (S’) are satisfied.

(1) Let (A) be valid. Then for the processes defined by (X3) and (Z3) one has

X_{3n}(ω_{1}) →^{d} W_{fvv′}, n → ∞ in C_{b}(R_{+})

for almost all ω_{1} ∈ Ω_{1}, where W_{fvv′}(x) = W (v^{2}f(x)) + W′(v^{′2}(1 − f(x))), x ∈ R^{+}.

(2) Let (A1) be valid. Then for the processes defined by (X3) and (Z3) one has

X_{3n}(ω_{1}) →^{d} W_{f_{m}vv′}, n → ∞ in C_{b}(R_{+})

for almost all ω_{1} ∈ Ω_{1}, where W_{f_{m}vv′}(x) = W (v^{2}f_{m}(x)) + W′(v^{′2}(1 − f_{m}(x))), x ∈ R^{+}.

Theorem 6. Suppose that (Y), (C), (Y’) and (C’) are satisfied.

(1) Let (A) be valid. Then for the processes defined by (X3) and (Z3’) one has

X′_{3n}(ω_{1}) →^{d} W_{fDD′}, n → ∞ in D_{b}(R_{+})

for almost all ω_{1} ∈ Ω_{1}, where W_{fDD′}(x) = W_{D}(f(x)) + W_{D′}((1 − f(x))), x ∈ R^{+}.

(2) Let (A1) be valid. Then for the processes defined by (X3) and (Z3’) one has

X′_{3n}(ω_{1}) →^{d} W_{f_{m}DD′}, n → ∞ in D_{b}(R_{+})

for almost all ω_{1} ∈ Ω_{1}, where W_{f_{m}DD′}(x) = W_{D}(f(x)) + W_{D′}((1 − f_{m}(x))), x ∈ R^{+}.

We will illustrate our theorems by following examples. Let 0 < α, m < ∞. Example 1. Let P_{1}(A_{ij}(n)) = p_{j}(n) = (1 − \frac{m}{k_{n}} )^{α}, for all i, j ∈ N. Then

for all x ∈ R_{+} it holds

\frac{E_{1}f_{1n}(x)}{k_{n}} = (1 − \frac{m}{k_{n}} )^{[xk_{n}]α} → e^{−mαx}, as n → ∞.

So we have:

If (Y) and (C), are satisfied, then by Theorem 1, for the processes defined by (X1) and (Z1’) one has X′_{1n}(ω) →^{d} W_{1D}, n → ∞ in D(R_{+}) for almost all ω_{1} ∈ Ω_{1}, where W_{1DD′}(x) = W_{D}(e^{−mαx}), x ∈ R_{+}.

If (Y), (Y’), (S), and (S’) are satisfied, then by Theorem 5 for the processes defined by (X3) and (Z3) one has X_{3n}(ω_{1}) →^{d} W_{1vv′}, n → ∞ in C_{b}(R_{+}) for almost all ω_{1} ∈ Ω_{1}, where W_{1vv′}(x) = W (v^{2}e^{−mαx}) + W′(v^{′2}(1 − e^{−mαx})), x ∈ R_{+}.

Example 2. Let P_{1}(A_{ij}(n)) = p_{j}(n) = (1 − \frac{m}{k_{n}+(j−1)m})^{α} for all i, j ∈ N (k_{n} > m). Then for all x ∈ R_{+} it holds

\frac{E_{1}f_{1n}(x)}{k_{n}} = (\frac{k_{n} + ([k_{n}x] − 2)m}{k_{n} + ([k_{n}x] − 1)m})^{α} (\frac{k_{n} + ([k_{n}x] − 3)m}{k_{n} + ([k_{n}x] − 2)m})^{α} … (\frac{k_{n} − m}{k_{n}})^{α}

= (\frac{k_{n} − m}{k_{n} + ([k_{n}x] − 1)m})^{α} → (\frac{1}{1 + mx})^{α} , as n → ∞.

If (Y) and (C), are satisfied, then by Theorem 2 for the processes defined by (X1) and (Z1’) one has X′_{1n}(ω_{1}) →^{d} W_{2D}, n → ∞, in D_{b}(R_{+}) for almost all ω_{1} ∈ Ω_{1}, where W_{2D}(x) = W_{D}(\frac{1}{(1+mx)^{α}}), x ∈ R_{+}.

If (Y), (Y’), (S), and (S’) are satisfied, then by Theorem 5, for the processes defined by (X3) and (Z3) one has X_{3n}(ω_{1}) →^{d} W_{2σσ′}, n → ∞ in C_{b}(R^{+}) for almost all ω_{1} ∈ Ω_{1}, where W_{2vv′}(x) = W (\frac{v^{2}}{(1+mx)^{α}}) + (W′(v^{′2}(1 − \frac{1}{(1+mx)^{α}})), x ∈ R_{+}.

Example 3. Let P_{1}(A_{ij}(n)) = p_{j}(n) = (1 − \frac{m}{k_{n}−(j−1)m})^{α} if k_{n} > (j − 1)m and P_{1}(A_{ij}(n)) = 0 if k_{n} ≤ (j − 1)m, i, j ∈ N. Let 0 ≤ x < \frac{1}{m}. Then we obtain

\frac{E_{1}f_{1n}(x)}{k_{n}} = (\frac{k_{n} − [k_{n}x]m}{k_{n} − ([k_{n}x] − 1)m})^{α} (\frac{k_{n} − ([k_{n}x] − 1)m}{k_{n} − ([k_{n}x] − 2)m})^{α} … (\frac{k_{n} − m}{k_{n}})^{α}

= (\frac{k_{n} − [k_{n}x]m}{k_{n}})^{α} → (1 − mx)^{α} , n → ∞, x < \frac{1}{m}.

If (Y) and (C) are satisfied, then by Theorem 1, for the processes defined by (X1) and (Z1’) one has X′_{1n}(ω_{1}) →^{d} W_{3D}, n → ∞, in D(R_{+}) for almost all ω_{1} ∈ Ω_{1}, where W_{3D}(x) = W_{D}((1 − mx)^{α}I_{[0, \frac{1}{m})}), x ∈ R_{+}.

If (Y), (Y’), (S), and (S’) are satisfied, then by Theorem 5, for the processes defined by (X3) and (Z3) one has X_{3n}(ω_{1}) →^{d} W_{3vv′}, n → ∞ in C_{b}(R_{+}) for almost all ω_{1} ∈ Ω_{1}, where W_{3vv′}(x) = W (v^{2}(1 − mx)^{α}I_{[0, \frac{1}{m})}) + W′(v^{′2}(1 − (1 − mx)^{α}I_{[0, \frac{1}{m})})) x ∈ R_{+}.

Let α = 1. Let us speak of the sense of Example 1 and Example 3. Example 1 corresponds to the case when at the l-th step we replace m summands in the sums S_{l−1}^{(1n)} by zero regardless of whether a summand equals zero or not.

Under Example 3, after each replacement, the number of Y_{ni} in S_{l−1}^{(1n)} decreases by about m elements. So Example 3 corresponds to the case when at the l-th step we replace about m nonzero summands in S_{l−1}^{(1n)} by zero.

The same is true for S_{l}^{(3n)} . In this case, we replace summands of S_{l−1}^{(1n)} by Y′_{ni}.

Now we will prove functional limit theorems for integrals of X′_{1n}, X′_{2n} and X′_{3n} which we will use in applications to models of a financial market. More precisely we will consider the random processes:

(Wi) Z_{in}(x) ≡ Z_{in}(x; ω_{1})(Y) = ∫_{0}^{t} X′_{in}(x)dx = \frac{1}{k_{n}} ∑_{j=0}^{[k_{n}x]−1} S_{j}^{(in)}+ \frac{\{k_{n}x\}}{k_{n}}S_{[k_{n}x]}^{(in)},

x ∈ IR_{+}, n ∈ IN, ω_{1} ∈ Ω_{1} and the random processes

(Wi’)

Z′_{in}(x) ≡ Z′_{in}(x; ω_{1})(Y) = \frac{1}{k_{n}} ∑_{j=0}^{[k_{n}x]} S_{j}^{(in)}, x ∈ IR_{+} , n ∈ IN , ω_{1} ∈ Ω_{1} ,

where i ∈ N.

Theorem 7. Assume that the conditions (Y) and (S) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (W1) weakly converge in C[0, T] as follows

Z_{1n}(ω_{1}) ⟶^{d} W_{1}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}, where W_{1} is a gaussian random process with the covariance B_{1}(t_{1}, t_{2}) = v^{2} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} f(max(x, y))dydx, t_{1}, t_{2} ∈ [0, T].

(2) Let (A1) be valid. Then the processes defined by (W1) weakly converge in C[0, T] as follows

Z_{1n}(ω_{1}) ⟶^{d} W_{m1}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}, where W_{m1} is a gaussian random process with the covariance B_{m1}(t_{1}, t_{2}) = v^{2} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} e^{−m max(x,y)}dxdy, t_{1}, t_{2} ∈ [0, T].

Theorem 8. Assume that the conditions (Y), and (S) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (W2) weakly converge in C[0, T] as follows

Z_{2n}(ω_{1}) ⟶^{d} W_{2}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}, where W_{2} is a gaussian random process with the covariance B_{2}(t_{1}, t_{2}) = v^{′2} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}}(1 − f(min(x, y)))dydx, t_{1}, t_{2} ∈ [0, T].

(2) Let (A1) be valid. Then the processes defined by (W2) weakly converge in C[0, T] as follows

Z_{2n}(ω_{1}) ⟶^{d} W_{m2}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}, where W_{m2} is a gaussian random process with the covariance B_{m2}(t_{1}, t_{2}) = v^{′2} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}}(1 − e^{−m min(x,y)})dydx, t_{1}, t_{2} ∈ [0, T].

Remark 1. By changing of the integration order we obtain more simple forms of the variance and the covariance function of W_{i}, i ∈ {1, 2}. The variance of W_{1} is equal to

v_{1}(t) = B_{1}(t, t) = v^{2}(∫_{0}^{t} f(x)xdx + ∫_{0}^{t} dx ∫_{x}^{t} f(y)dy) = 2v^{2} ∫_{0}^{t} f(x)xdx,

the covariance of W_{1} is equal to

B_{1}(t_{1}, t_{2}) = v^{2} (∫_{0}^{t_{1}} f(x)xdx + ∫_{0}^{t_{1}} dx ∫_{x}^{t_{2}} f(y)dy) =

v^{2}(t_{1} ∫_{t_{1}}^{t_{2}} f(x)dx + 2 ∫_{0}^{t_{1}} f(x)xdx), 0 ≤ t_{1} ≤ t_{2}.

The variance of W_{2} is equal to

v_{2}(t) = B_{1}(t, t) = v^{2}(t^{2} − 2t ∫_{0}^{t} f(x)dx + 2 ∫_{0}^{t} f(x)xdx),

the covariance of W_{2} is equal to

B_{2}(t_{1}, t_{2}) = v^{2}(t_{1}t_{2} − (t_{1} + t_{2}) ∫_{0}^{t_{1}} f(x)dx + 2 ∫_{0}^{t_{1}} f(x)xdx), 0 ≤ t_{1} ≤ t_{2}.

Theorem 9. Assume that (Y), (Y’) ,(S) and (S’) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (W3) weakly converge in C[0, T] as follows

Z_{3n}(ω_{1}) ⟶^{d} W_{3}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}, where W_{3} is a gaussian random process defined on [0, T] with the following covariance function under t_{1}, t_{2} ∈ [0, T], B_{3}(t_{1}, t_{2}) = B_{1}(t_{1}, t_{2}) + B_{2}(t_{1}, t_{2}).

(1) Let (A1) be valid. Then the processes defined by (W3) weakly converge in C[0, T] as follows

Z_{3n}(ω_{1}) ⟶^{d} W_{3}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}, where W_{3} is a gaussian random process defined on [0, T] with the following covariance function under t_{1}, t_{2} ∈ [0, T], B_{m3}(t_{1}, t_{2}) = B_{m1}(t_{1}, t_{2}) + B_{m2}(t_{1}, t_{2}).

Proof. Observe that

Z_{3n} = Z_{1n}(Y) + Z_{2n}(Y′), n ∈ N,

and Z_{1n}(Y) and Z_{2n}(Y′) are independent random processes. So Theorem 9 is following from Theorem 7 and Theorem 8.

Theorem 10. Assume that the conditions (Y), and (S) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (W1’) weakly converge in L^{∞}[0, T] as follows

Z′_{1n}(ω_{1}) ⟶^{d} W_{1}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

(2) Let (A1) be valid. Then the processes defined by (W1’) weakly converge in L^{∞}[0, T] as follows

Z′_{1n}(ω_{1}) ⟶^{d} W_{m1}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

Theorem 11. Assume that the conditions (Y), and (S) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (W2’) weakly converge in L^{∞}[0, T] as follows

Z′_{2n}(ω_{1}) ⟶^{d} W_{2}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}

(2) Let (A2) be valid. Then the processes defined by (W2’) weakly converge in L^{∞}[0, T] as follows

Z′_{2n}(ω_{1}) ⟶^{d} W_{m2}, as n → ∞

for almost all ω_{1} ∈ Ω_{1} Theorem 10 and Theorem 11 follow from Theorem G.

Theorem 12. Assume that (Y), (Y’) ,(S) and (S’) are fulfilled.

(1) Let (A) be valid. Then the processes defined by (W3’) weakly converge in L^{∞}[0, T] as follows

Z′_{3n}(ω_{1}) ⟶^{d} W_{3}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

(1) Let (A1) be valid. Then the processes defined by (W3’) weakly converge in L^{∞}[0, T] as follows

Z′_{3n}(ω_{1}) ⟶^{d} W_{m3}, as n → ∞

for almost all ω_{1} ∈ Ω_{1}.

Proof. Observe that Z′_{3n} = Z′_{1n}(Y) + Z′_{2n}(Y′), n ∈ N, and Z′_{1n}(Y) and Z′_{2n}(Y′) are independent random processes. So Theorem 12 follows from Theorem 10 and Theorem 11.

Remark 2. Since trajectories of Z′_{in} are random step lines with jumps in rational points, in Theorem 10, Theorem 11 and Theorem 12 instead L^{∞}[0, T] we can consider closed subspace B of L^{∞}[0, T], containing such step lines. This space is separable. Therefore the class of the functionals h : B → R_{+} such that Eh(Z_{in}) → Eh(W_{i}) as n → ∞, is sufficiently wide.

Remark 3. We consider the case when 1I_{ji}(n), Y_{ni} and Y′_{ni} are defined on the same probability space (Ω, 𝔄, P) and for all n ∈ N, 1I_{ji}(n), Y_{ni}, Y′_{ni}, i ∈ N are independent random variables. In this case, the random processes X_{in}, X′_{in}, Z_{in}, and Z′_{in} will be denoted by X_{(i+3)n}, X′_{(i+3)n}, Z_{(i+3)n}, and Z′_{(i+3)n}.

Then Theorem 1 - Theorem 12 hold in the case when instead X_{in}, X′_{in}, Z_{in}, and Z′_{in}, we use X_{(i+3)n}, X′_{(i+3)n}, Z_{(i+3)n}, and Z′_{(i+3)n}, respectively. Of course, in this case the phrase “for almost all ω_{1} ∈ Ω_{1}” should be omitted. This follows from the following assertion.

Let V_{n}(ω_{1}), V : Ω → B, n ∈ N be random elements in a separable metrizable space B with the parameter ω_{1} ∈ Ω_{1} which are Bochner measurable as functions defined on Ω × Ω_{1} and such that V_{n}(ω_{1}) →^{d} V, n → ∞ for almost all ω_{1} ∈ Ω_{1}. Define random elements V′_{n} by L(V′_{n})(A) = E_{1}L(V_{n})(ω_{1})(A), where A is a Borel subset of B. Then V′_{n} →^{d} V, n → ∞.

Proof. Let f : B → R be a continuous bounded function. Then for almost all ω_{1} ∈ Ω_{1} it holds Ef(V_{n}(ω_{1})) → Ef(V), n → ∞. So by Lebesgue Theorem, E_{1}Ef(V_{n}) → Ef(V), as n → ∞. Observe that E_{1}Ef(V_{n}) = Ef(V′_{n}), n ∈ N. Therefore Ef(V′_{n}) → Ef(V), n → ∞ and. V′_{n} →^{d} V, as n → ∞ in B.

Remark 4. We will suppose that I_{0i}(n) = 1, i ∈ N. Consider the integer valued random variables defined as

N_{1n}(l) = ∑_{i=1}^{k_{n}} 1I_{li}(n)1I_{(l−1)i}(n) ⋯ 1I_{1i}(n), N_{2n}(l) = k_{n} − N_{1n}(l), l, n ∈ N.

Next,

S_{l}^{((6+i)n)} = ∑_{i=1}^{N_{jn}(l)} Y_{ni}, i ∈ {1, 2}, S_{l}^{(9n)} = ∑_{i=1}^{N_{1n}(l)} Y_{ni} + ∑_{i=1}^{N_{2n}(l)} Y′_{ni}, l ∈ N.

Then for the processes defined by (Z(i + 6)′), (Z(i + 6)), (W (i + 6)) and (W (i + 6)′) for i ∈ {1, 2, 3} we have: L(X′_{(i+3)n}) = L(X′_{(i+6)n}), L(X_{(i+3)n}) = L(X_{(i+6)n}), L(Z′_{(i+3)n}) = L(Z′_{(i+6)n}) and L(Z_{(i+3)n}) = L(Z_{i+6)n}). The sums S_{l}^{((i+6)n)} are sums with random index of summing (random sums). Thus, Remark 2 may be considered as a functional limit theorem for random sums. See, for example, Korolev and Kruglov [2] for limit theorems for random sums.

5. Applications to models of market pricing

Consider the market model with n agents, where n is large enough. At any discrete time t ∈ {0, 1, … M} = T each agent buys or sells asset which is called below a stock (which is called agents operation).

An agent operation in the market may take place in the moments j ∈ {0, 1, 2 … M} only. And operations consists that an agent buys or sells an asset which is called below a stock. At the each moment j of the time i-th agent can buy a stock with the probability p and sell the stock with the probability q = 1 − p. If an agent buy a stock then the stock price is multiplied by the number U, U > 1. If an agent sell a stock then the stock price is multiplied by the number V, 0 < V < 1. So we suppose that the trading policy of an i-th agent at the moment j ∈ T is a random variable with the distribution

ξ_{ij} = \{ \begin{matrix}U & with probability & p \\ D & with probability & q \end{matrix} , 1 ≤ i ≤ n, 0 ≤ j ≤ m.

Thus, the random variable ξ_{ij} is a trade policy of the i-th agent in the moment j.

Introduce the parameter a and v > 0 by the following equalities

\begin{matrix}v^{2} & = & (U − V)^{2}pq \\ av^{2} − 1 & = & Up + Dq \end{matrix} \} .

It is easy to see that v^{2} = σ^{2}(ξ_{ij}). By analogy to the classical limit transition we assume that σ^{2}(ξ_{ji}) = \frac{σ_{0}^{2}}{n}, where σ_{0}^{2} is a volatility constant not depending from n. The parameter a named as relative risk characterizes difference between ”market” measure (p, q) and ”risk-neutral” measure (p̃, q̃), i.e. the numbers p̃ and q̃ are define by Up̃ + Dq̃ = 1, q̃ = 1 − q̃. Correspondingly to the classical case we assume that

a = a(n) → \frac{μ − r}{σ_{0}^{2}} as, n → ∞, μ − r ∈ R_{+}, r > 0,

where μ is an expected rate of stock return and r is an interest rate of discount bond: B_{t} = exp{rt}.

Next we assume that every agent makes equivalent contribution in to pricing of stock S by his trading policy.

(G) At any fixed moment k ∈ T, each agent’s contribution in to pricing is defined as an evolution of his trading policy during the time interval {0, 1, … k}

Z_{k}(i) = ∏_{j=0}^{k} ξ_{ij}, k = 0, 1, …, m.

The following assumption is more important in the construction of the mode. (F) At any moment k ∈ T the value of stock is equal to geometric mean of

contribution in to pricing over n agents

S_{k} = S_{0} (∏_{i=1}^{n} Z_{k}(i))^{1/n} = S_{0} (∏_{i=1}^{n} ∏_{j=0}^{k} ξ_{ij})^{1/n} ,

where S_{0} is a starting point.

In accordance to the assumption of time parameterizations and to the assumption (G) for the order of the model n we define the value of the stock at the time moment t ∈ [0, T] by

S_{n}(t) = S_{0} (∏_{i=1}^{n} Z_{[nt]}(i))^{1/n} = S_{0} (∏_{i=1}^{n} ∏_{j=0}^{[nt]} ξ_{ij})^{1/n} .

So by this equality we have defined a family of step lines with jumps in rational points, depending from the random parameter ω_{1} ∈ Ω_{1} which we consider as elements in L^{∞}[0, T].

The market probability measure (p, q) induces the random process S_{n} = S_{n}(p, q). Analogously, the risk-neutral measure (p̃, q̃), induces the random process S_{n} = S_{n}(p̃, q̃). Recall that the rational value of the Standard European Call Option C_{T} with the maturity T for a strike K can be defined as

C_{T} = E{e^{−rT} (e^{rT} S̃(T) − K)_{+}},

where the random process S(t), t ∈ [0, T] is a limit of S_{n}(p̃, q̃) as n → ∞.

Model 1. Changing of a trading policy. We will assume that our market satisfies to the following property:

(A3) Actions of each agent does not depends from actions of another agents and trading policy evaluates by following: if the indicates I_{i1} = 1, I_{i2} = 1, ⋯ I_{ij} = 1, then the trade policy of i-th agent is remaining the same and is defining by the random variable ξ_{i}: ξ_{ik} = ξ_{i}, k ≤ j; If the indicators I_{i1} = 1, I_{i2} = 1, ⋯ I_{i(j−1)} = 1, but the indicator I_{ij} = 0, then ξ_{i} is replaced by independent copy ξ′_{i}: ξ_{ik} = ξ′_{i}, k ≥ j. So the evolution consists that in a random moment of a trade policy is changing. That is ξ′_{i} replaces the random variable ξ_{i}. This replacement can take place only one time.

Theorem 13. (The case of the market probability.)(1) Under assumption (A), (A2), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p, q) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S(t) = S_{0} exp\{(μ−r)t+W_{3}(t)−\frac{σ_{0}^{2}}{2} t\}, t ∈ [0, T] and W_{3} is a gaussian random process with the covariance function B_{3}(t_{1}, t_{2}) = σ_{0}^{2}(B_{1}(t_{1}, t_{2}) + B_{2}(t_{1}, t_{2})), t_{1}, t_{2} ∈ [0, T].

(2) Under assumption (A1), (A2), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p, q) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S(t) = S_{0} exp\{(μ−r)t+W_{3}(t)−\frac{σ_{0}^{2}}{2} t\}, t ∈ [0, T] and W_{3} is a gaussian random process with the covariance function B_{m3}(t_{1}, t_{2}) = σ_{0}^{2}(B_{m1}(t_{1}, t_{2}) + B_{m2}(t_{1}, t_{2})), t_{1}, t_{2} ∈ [0, T].

Theorem 14. (The case of the risk-neutral probability.) (1) Under assumption (A), (A2), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p̃, q̃) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S̃(t) = S_{0} exp\{(W_{3}(t) − \frac{σ_{0}^{2}}{2} t\} t ∈ [0, T] .

(2) Under assumption (A1), (A2), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p̃, q̃) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S̃(t) = S_{0} exp\{(W_{3}(t) − \frac{σ_{0}^{2}}{2} t\} t ∈ [0, T].

Theorem 15. (The Black-Scholes Formula for the model.) Let (A) or (A1), (A2), (G), (F), (X3) and (Z3) be valid. Denote the variance

v^{2}(t) = E(W_{3}(t))^{2} = σ_{0}^{2} (∫_{0}^{t} ∫_{0}^{t}(1 − f(min(x, y)) + f(max(x, y)))dxdy) ,

where t ∈ [0, T], in the case (A), and

v^{2}(t) = E(W_{3}(t))^{2} = σ_{0}^{2} (∫_{0}^{t} ∫_{0}^{t}(1 − e^{−min(x,y)} + e^{− max(x,y)})dxdy) , t ∈ [0, T]. in the case (A1).

Then the rational value of the Standard European Call Option is equal to

C_{T} = S_{0} exp\{\frac{v^{2}(T)}{2} − \frac{σ_{0}^{2}}{2} T \}Φ(ρ + v(T)) − K exp\{−rT \}Φ(ρ),

where

ρ = \frac{ln S_{0} − ln K + rT − σ_{0}^{2}T/2}{v(T)}.

Model 2. Decreasing market. We consider the case of a decreasing market, that is agents go out from a market. In this case the random variables ξ_{i} are replaced by ξ′_{i} = 1 and (an agent which gone out from market does not can to influence on a stock price) and ln(ξ_{i}) is replacing by zeros. We consider a leaving of agents from market as a replacement by zeros. So will assume that it is valid the condition:

(A3) Actions of each agent does not depends from actions of another agents and trading policy evaluates by following: if the indicates I_{i1} = 1, I_{i2} = 1, ⋯ I_{ij} = 1, then the trade policy of i-th agent is remaining the same and is defining by the random variable ξ_{i}: ξ_{ik} = ξ_{i}, k ≤ j; If the indicators I_{i1} = 1, I_{i2} = 1, ⋯ I_{i(j−1)} = 1, but the indicator I_{ij} = 0, then ξ_{i} is replaced by one: ξ_{ik} = 1 for k ≥ j. That is at the moment j ith agent go out from a market. And we have the following analogs of Theorem 13, Theorem 14 and Theorem 15.

Theorem 16. (The case of the market probability.) (1) Under assumption (A), (A3), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p, q) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S(t) = S_{0} exp{(μ − r)b(t) + W_{1}(t) −

\frac{σ_{0}^{2}}{2} b(t)\}, t ∈ [0, T] and W_{1} is a gaussian random process with the covariance function

B_{1}(t_{1}, t_{2}) = σ_{0}^{2} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} f(max(x, y))dxdy, t_{1}, t_{2} ∈ [0, T]

and

b(t) = ∫_{0}^{t} f(x)dx, t ∈ [0, T].

(2) Under assumption (A1) (A3), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p, q) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S(t) = S_{0} exp{(μ − r)b(t) + W_{1}(t) −

\frac{σ_{0}^{2}}{2} b(t)\}, t ∈ [0, T] and W_{1} is a gaussian random process with the covariance function

B_{m1}(t_{1}, t_{2}) = σ_{0}^{2} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} e^{−m max(x,y)}dxdy, t_{1}, t_{2} ∈ [0, T]

and

b(t) = ∫_{0}^{t} e^{−mx}dx, t ∈ [0, T].

Theorem 17. (The case of the risk-neutral probability.) Under assumption (A) or (A1), (A3), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p̃, q̃) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S̃(t) = S_{0} exp\{W_{1}(t) − \frac{σ_{0}^{2}}{2} b(t)\} t ∈ [0, T], and in the case (A) W_{1}(t) and b(t) are defined in (1) of Theorem 16 and in the case (A1) W_{1}(t) and b(t) are defined in (2) of Theorem 16.

Theorem 18. (The Black-Scholes Formula for the model.) Let (A) or (A1), (A3), (G) and (F) be valid. Denote

v^{2}(t) = E(W_{1}(t))^{2} = B_{1}(t, t) = 2σ_{0}^{2} ∫_{0}^{t} ∫_{0}^{t} f(max(x, y))dxdy,
b(t) = ∫_{0}^{t} f(x)dx

in the case (A), and

v^{2}(t) = E(W_{1}(t))^{2} = B_{1}(t, t) = 2σ_{0}^{2} ∫_{0}^{t} ∫_{0}^{t} e^{−m max(x,y)}dxdy,
b(t) = ∫_{0}^{t} e^{−mx}dx

in the case (A1).

Then the rational value of the Standard European Call Option is equal to

C_{T} = S_{0} exp\{\frac{v^{2}(T)}{2} − \frac{σ_{0}^{2}}{2} b(T)\}Φ(ρ + v(T)) − K exp\{−rb(T)\}Φ(ρ),

where

ρ = \frac{ln S_{0} − ln K + rb(T) − σ_{0}^{2}b(T)/2}{v(T)}.

Model 3. Increasing Market. We consider the case of an increasing market, that is agents come to a market. In this case the random variables ξ_{i} = 1 is replacing by ξ′_{i}: an agents which come to a market begin to influence on a stock price. And we will use the following condition

(A4) Actions of each agent does not depends from actions of another agents and trading policy evaluates by following: if the indicates I_{i1} = 1, I_{i2} = 1, ⋯ I_{ij} = 1, then the trade policy of i-th agent is equal to one: ξ_{ik} = 1, k ≤ j; If the indicators I_{i1} = 1, I_{i2} = 1, ⋯ I_{i(j−1)} = 1, but the indicator I_{ij} = 0, then ξ_{ik} = ξ_{i}, k ≥ j. This means that at the jth moment ith agent comes to a market.

As analogs of Theorem 13, Theorem 14 and Theorem 15 we have Theorem 19. (The case of the market probability.) (1) Under assumption

(A), (A4), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p, q) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S(t) = S_{0} exp{(μ − r)b(t) + W_{2}(t) −

\frac{σ_{0}^{2}}{2} b(t)\}, t ∈ [0, T] and W_{2} is a gaussian random process with the covariance function

B_{2}(t_{1}, t_{2}) = σ_{0}^{2} (t_{1}t_{2} − ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} f(min(x, y))dxdy) , t_{1}, t_{2} ∈ [0, T],

and

b(t) = t − ∫_{0}^{t} f(x)dx.

(2) Under assumption (A1), (A4), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p, q) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S(t) = S_{0} exp{(μ − r)b(t) + W_{2}(t) −

\frac{σ_{0}^{2}}{2} b(t)\}, t ∈ [0, T] and W_{2} is a gaussian random process with the covariance function

B_{m2}(t_{1}, t_{2}) = σ_{0}^{2} (t_{1}t_{2} − ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} e^{−m min(x,y)}dxdy) t_{1}, t_{2} ∈ [0, T]

and

b(t) = t − ∫_{0}^{t} e^{−mx}dx.

Theorem 20. (The case of the risk-neutral probability.)Under assumption (A) or (A1, (A4), (G) and (F) the following convergence is valid as n → ∞

S_{n}(p̃, q̃) →^{d} S

in L^{∞}[0, T] for almost all ω_{1} ∈ Ω_{1}, where S̃(t) = S_{0} exp\{W_{2}(t) − \frac{σ_{0}^{2}}{2} b(t)\} t ∈ [0, T] and in the case (A) W_{2}(t) and b(t) are defined in (1) of Theorem 19 and in the case (A1) W_{2}(t) and b(t) are defined in (2) of Theorem 19.

Theorem 21. (The Black-Scholes Formula for the model.) Let (A) or (A1), (A4), (G) and (F) be valid. Denote

v^{2}(t) = E(W_{2}(t))^{2} = σ_{0}^{2} (t^{2} − ∫_{0}^{t} ∫_{0}^{t} f(min(x, y))dydx) ,
b(t) = t − ∫_{0}^{t} f(x)dx.

in the case (A), and

v^{2}(t) = E(W_{2}(t))^{2} = σ_{0}^{2} (t^{2} − ∫_{0}^{t} ∫_{0}^{t} e^{−m min(x,y)}dydx) ,
b(t) = t − ∫_{0}^{t} e^{−mx}dx.

in the case (A2).

Then the rational value of the Standard European Call Option option is equal to

C_{T} = S_{0} exp\{\frac{v^{2}(T)}{2} − \frac{σ_{0}^{2}}{2} b(T)\}Φ(ρ + v(T)) − K exp\{−rb(T)\}Φ(ρ),

where ρ = \frac{ln S_{0}−ln K+rb(T)−σ_{0}^{2}b(T)/2}{v(T)}.

The proofs of Theorem 13 – Theorem 21 are the same as the proofs of Theorem 1, Theorem 2 and Theorem 3 [9] (see, also, [10]) with the difference that is instead of Theorem 1 from [6] we use Theorem 10, Theorem 11 and Theorem 12 respectively.

REFERENCES

[1] Kolchin,V.F., Sevast’ianov,B.A., Chistiakov,V.P.) (1976) Random occupations. M.:Nauka. 1976 (in Russian).

[2] Kruglov V. M. and Koroglev V. Iu. (1990) Limit theorems for random sums. Moscow University Press, Moscow (in Russian).

[3] Prokhorov, Yu. V. (1956). Convergence of random processes and limit theorems of theory of probability. Theory of Probability and Its Applications. 1(2), 117 - 238 (in Russian).

[4] Gihman I.I., and Skorokhod A.V. (1977).Introduction to the theory of stochastic processes. Nauka, Moscow (in Russian).

[5] Fazekas, I., Chuprunov, A. (1996). Convergence of random step lines to Ornstein–Uhlenbeck type processes. Technical Report of the Debrecen University, No. 24/1996.

[6] Rusakov, O.V. (1995). The functional limit theorem for random variables with strong remaining dependence. Theory of Probability and Its Applications. 40 813 - 832 (in Russian).

[7] Loév, M. (1960).Probability theory. D. van Nostand company inc. , Princeton - New Jersey - Toronto, New-York - London.

[8] Billingsley P. (1968) Convergence of probability measures. John Wiley & Sons., New York - London - Sydney - Toronto.

[9] Rusakov O.V. (1998) A model of market pricing with randomly distributed information and the geometric integral of the Ornstein-Uhlenbeck process. Report of the Department of Mathematics. University of Helsinki. Preprint 184. April 1998. 13 p.

[10] Rusakov O.V. (1994) Asymptotic behavior of sums of random variables with random replacements. Zapiski nauchnih seminarov POMI. St.Petersburg. V. 216, 714–728.

Kazan State University, Chebotarev Institute of Mathematics and Mechanics, Universitetskaia 17, 420008 Kazan, Russia E-mail address: Alexey.Chuprunov@ksu.ru St.Petersburg State University, Department of Mathematics and Mechanics, 198904, Bibliotechnaya Square 2, Starii Petergoff, Russia E-mail address: Oleg.Rusakov@pobox.spbu.ru

Received February 12, 2003