Lobachevskii Journal of Mathematics

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Vol. 15, 2004, 21 – 33 \begin{matrix}c \\ ◯ \end{matrix} A. Volodin, R. Giuliano Antonini, T.-C. Hu

A. Volodin, R. Giuliano Antonini, and T.-C. Hu

A NOTE ON THE RATE OF COMPLETE CONVERGENCE

FOR WEIGHTED SUMS OF ARRAYS OF BANACH SPACE VALUED RANDOM ELEMENTS

(submitted by D. Mushtari)

Abstract. We obtain complete convergence results for arrays of rowwise independent Banach space valued random elements. In the main result no assumptions are made concerning the geometry of the underlying Banach space. As corollaries we obtain results on complete convergence in stable type p Banach spaces.

1. Introduction.

The concept of complete convergence was introduced by Hsu and Robbins (1947) as follows. A sequence of random variables {U_{n}, n ≥ 1} is said to converge completely to a constant c if ∑_{n=1}^{∞} P {|U_{n} − c| > ε} < ∞ for all ε > 0. By the Borel-Cantelli lemma, this implies U_{n} → c almost surely (a.s.) and the converse implication is true if the {U_{n}, n ≥ 1} are independent. Hsu and Robbins (1947) proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite.

probability.

2000 Mathematical Subject Classification. 60B12, 60F05, 60F25, 60G42.

Key words and phrases. array of Banach space valued random elements, stable type p Banach space, rowwise independence, weighted sums, complete convergence, rate of convergence, almost sure convergence, convergence in

This result has been generalized and extended in several directions (see Pruitt (1966), Rohatgi (1971), Hu, Moricz, and Taylor (1989), Gut (1992), Wang, Rao, and Yang (1993), Kuczmaszewska and Szynal (1994), Sung (1997), Hu, Rosalsky, Szynal, and Volodin (1999), Hu, Li, Rosalsky and Volodin (2002), and Ahmed, Giuliano Antonini, and Volodin (2002) among others). Some of these articles concern a Banach space setting. A sequence of Banach space valued random elements is said to converge completely to the 0 element of the Banach space if the corresponding sequence of norms converges completely to 0.

Hu, Rosalsky, Szynal, and Volodin (1999) presented a general result (cf. Theorem 1.1 below) establishing complete convergence for the row sums of an array of rowwise independent but not necessarily identically distributed Banach space valued random elements. Their result also specified the corresponding rate of convergence. The Hu, Rosalsky, Szynal, and Volodin (1999) result unifies and extends previously obtained results in the literature in that many of them (for example, results of Hsu and Robbins (1947), Hu, Li, Rosalsky, and Volodin (2002), Hu, Moricz, and Taylor (1989), Gut (1992), Kuczmaszewska and Szynal (1994), Pruitt (1966), Rohatgi (1971), Sung (1997), and Wang, Rao, and Yang (1993)) follow from it.

In the following we assume that {V_{nk}, k ≥ 1, n ≥ 1} is an array of rowwise independent random elements in a separable real Banach space and {a_{nk}, k ≥ 1, n ≥ 1} is an array of constants. Denote

S_{n} ≡ ∑_{k=1}^{∞} a_{nk}V_{nk}.

In the next theorem the weights a_{nk} are built into the array (that is, a_{nk} = 1 for all k and n).

Theorem 1.1 (Hu, Rosalsky, Szynal, and Volodin (1999)). Let {c_{n}, n ≥ 1} be a sequence of positive constants. Suppose that

∑_{n=1}^{∞} c_{n} ∑_{k=1}^{∞} P {||V_{nk}|| > ε} < ∞ for all ε > 0, (1)

(2) ∑_{n=1}^{∞} c_{n} (∑_{k=1}^{∞} E||V_{nk}||^{q})^{J} < ∞ for some 0 < q ≤ 2 and J ≥ 2,

∑_{k=1}^{∞} V_{nk} →^{P} 0, and

A NOTE ON THE RATE OF COMPLETE CONVERGENCE 23

(3) if lim inf_{n→∞} c_{n} = 0, then ∑_{k=1}^{∞} P {||V_{nk}|| > δ} = o(1) for some δ > 0.

Then

∑_{n=1}^{∞} c_{n}P {||S_{n}|| > ε} < ∞ for all ε > 0.

It is implicitly assumed in Theorem 1.1 that the series S_{n} converges a.s.

The article Hu, Li, Rosalsky, and Volodin (2002) is devoted to presenting applications of Theorem 1.1 to obtain new complete convergence results. Theorem 1.2 generalizes results of Hsu and Robbins (1947), Hu, Moricz, and Taylor (1989), Gut (1992), Kuczmaszewska and Szynal (1994), Pruitt (1966), Rohatgi (1971), Sung (1997), and Wang, Rao, and Yang (1993) in three directions, namely:

(i) Banach space valued random elements instead of random variables are considered. (ii) An array rather than a sequence is considered. (iii) The rate of convergence is obtained.

Theorem 1.2. (Hu, Li, Rosalsky, and Volodin (2002)). Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} is stochastically dominated by a random variable X. Assume that

sup_{k≥1} |a_{nk}| = O(n^{−γ}) for some γ > 0, and

∑_{k=1}^{∞} |a_{nk}| = O(n^{α}) for some α ∈ [0, γ).

If

E|X|^{1+ \frac{1+α+β}{γ}} < ∞ for some β ∈ (−1, γ − α − 1], and

S_{n} →^{P} 0,

then

∑_{n=1}^{∞} n^{β}P {||S_{n}|| > ε} < ∞ for all ε > 0.

The proof of Theorem 1.2 is rather complicated once it uses the Stieltjes integral techniques, summation by parts lemma and so on. The initial objective of an investigation resulted in the paper Ahmed, Giuliano Antonini, and Volodin (2002) was only to find a simpler proof. But it appears that they were able to establish a more general result and with simpler proof. The result presented in Theorem 1.3 below is more general than the main result of Hu, Li, Rosalsky, and Volodin (2002), since rates of convergence for moving averages can be established, which cannot be proved using Theorem 1.2.

Theorem 1.3. Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} is stochastically dominated by a random variable X. Assume that

sup_{k≥1} |a_{nk}| = O(n^{−γ}) for some γ > 0,

and

∑_{k=1}^{∞} |a_{nk}| = O(n^{α}) for some α < γ.

Let β be such that α + β ≠ −1 and fix δ > 0 such that \frac{α}{γ} + 1 < δ ≤ 2. If

E|X|^{ν} < ∞ where ν = max(1 + \frac{1 + α + β}{γ} , δ),

and

S_{n} →^{P} 0,

then

∑_{n=1}^{∞} n^{β}P {||S_{n}|| > ε} < ∞ for all ε > 0.

If we assume that the Banach space has the geometric property of being of an appropriate stable type, then we can drop the condition that S_{n} converges in probability. For this we need the following result.

Theorem 1.4 (Adler, Ordóñez Cabrera, Rosalsky, and Volodin (1999)) Assume that underlying Banach space has stable type p, 1 < p < 2. Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} has mean 0 and is stochastically dominated by a random variable X. Moreover, assume that

sup_{n≥1} ∑_{k=1}^{∞} |a_{nk}|^{p} < ∞, and sup_{k≥1} |a_{nk}| = o(1).

If lim_{t→∞} t^{p}P {|X| > t} = 0, then S_{n} →^{P} 0.

In the present paper we generalize Theorem 1.3 in two directions. First of all, instead of condition

∑_{k=1}^{∞} |a_{nk}| = O(n^{α}) for some α < γ,

in Theorem 3.1 we consider the condition

∑_{k=1}^{∞} |a_{nk}|^{θ} = O(n^{α}) for some suitable choice of the parameters.

Secondary, in Theorem 3.2 we deal with the special case α + β = −1.

The plan of the paper is as follows. In Section 2, we recall some well known definitions pertaining to the current work. In Section 3, we apply Theorem 1.1 to obtain complete convergence for row sums with corresponding rates of convergence (Theorems 3.1 and 3.2). As in Theorems 1.1, 1.2, and 1.3, in Theorems 3.1 and 3.2 no assumptions are made concerning the geometry of the underlying Banach space. We use the geometrical assumption (type p) on the underlying Banach space in Theorems 3.3 and 3.4.

2. Preliminaries

Let (Ω, F, P) be a probability space and let B be a separable real Banach space with norm || · ||. A random element is defined to be an F -measurable mapping of Ω into B equipped with the Borel σ-algebra (that is, the σ-algebra generated by the open sets determined by || · ||). A detailed account of basic properties of random elements in separable real Banach spaces can be found in Taylor (1978).

Let {V_{nk}, k ≥ 1, n ≥ 1} be an array of rowwise independent, but not necessarily identically distributed, random elements taking values in B. The array of random elements {V_{nk}, k ≥ 1, n ≥ 1} is said to be stochastically dominated by a random variable X if there exists a constant D < ∞ such that P {||V_{nk}|| > x} ≤ DP {|DX| > x} for all x > 0 and for all n ≥ 1 and k ≥ 1. Let {a_{nk}, k ≥ 1, n ≥ 1} be an array of constants (called weights) and consider the sequence of weighted sums S_{n} ≡ ∑_{k=1}^{∞} a_{nk}V_{nk}, n ≥ 1.

Let 0 < p ≤ 2 and let {θ_{n}, n ≥ 1} be independent and identically distributed stable random variables each with characteristic function ϕ(t) = exp{−|t|^{p}}, −∞ < t < ∞. The real separable Banach space B is said to be of stable type p if ∑_{n=1}^{∞} θ_{n}v_{n} converges almost surely whenever {v_{n}, n ≥ 1} ⊆ B with ∑_{n=1}^{∞} ||v_{n}||^{p} < ∞. Equivalent characterizations of a Banach space being of stable type p, properties of stable type p Banach spaces, as well as various relationships between the conditions “Rademacher type p” and “stable type p” may be found in Adler, Ordóñez Cabrera, Rosalsky, and Volodin (1999), Section 2.

Finally, the symbol C denotes throughout a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance, for x ≥ 0 the symbol [x] denotes the greatest integer in x, and for a finite set A the symbol #A denotes the number of elements in the set A.

3. Main Results

With the preliminaries accounted for, the main result may now be established. For the case θ > 1 it is implicitly assumed that the series S_{n} converges a.s. (cf. Remark (i) after Theorem 3.2)

Theorem 3.1. Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} is stochastically dominated by a random variable X. Assume that

(4)  sup_{k≥1} |a_{nk}| = O(n^{−γ}) for some γ > 0, and

∑_{k=1}^{∞} |a_{nk}|^{θ} = O(n^{α}) (5)

for some 0 < θ ≤ 2 and any α such that θ + \frac{α}{γ} < 2.

Let β be such that α + β ≠ −1 and fix δ > θ such that \frac{α}{γ} + θ < δ ≤ 2. If

(6) E|X|^{ν} < ∞ where ν = max(θ + \frac{1 + α + β}{γ} , δ), and

S_{n} →^{P} 0,

then

∑_{n=1}^{∞} n^{β}P {||S_{n}|| > ε} < ∞ for all ε > 0.

Proof. Note that the result is of interest only for β ≥ −1. In addition, since δ > θ + \frac{α}{γ}, we have that α − γ(δ − θ) < 0.

Let c_{n} = n^{β}, n ≥ 1. Then we only need to verify that the conditions (1), (2), and (3) (if β < 0) of Theorem 1.1 hold with a_{nk}V_{nk} playing the role of V_{nk} in the formulation of that theorem. Without loss of generality, ε can be taken to be 1 and in view of (4) and (5) we can assume that

sup_{k≥1} |a_{nk}| = n^{−γ}, (7)

and

∑_{k=1}^{∞} |a_{nk}|^{θ} = n^{α}. (8)

Now to verify (1), let b_{nj} = 1/|a_{nj}|, ( where 1/0 = ∞),

I_{nj} = {k : (nj)^{γ} ≤ b_{nk} < (n(j + 1))^{γ}} and

J_{nj} = ∪_{k=1}^{j}I_{nk} = {k : b_{nk} < (n(j + 1))^{γ}}, j ≥ 1, n ≥ 1.

Noting that by (7) ∪_{j≥1}I_{nj} = N for all n ≥ 1, where N is the set of positive integers.

Note that for all m ≥ 1, n ≥ 1 by (8)

n^{α} ≥ ∑_{k∈J_{nm}} |a_{nk}|^{θ} = ∑_{k∈J_{nm}} 1/|b_{nk}| ≥ \frac{#J_{nm}}{(n(m + 1))^{γ}} ,

and hence

#J_{nm} ≤ n^{α+γθ}(m + 1)^{γθ}.

Observe that since the sets I_{nj}, j ≥ 1 are disjoint, for any fixed n ≥ 1, we have that

∑_{j=1}^{m} #I_{nj} = #J_{nm} ≤ n^{α+γθ}(m + 1)^{γθ}.

Therefore

∑_{n=1}^{∞} n^{β} ∑_{k=1}^{∞} P {∥a_{nk}V_{nk}∥ ≥ 1} = ∑_{n=1}^{∞} n^{β} ∑_{k=1}^{∞} P {∥V_{nk}∥ ≥ b_{nk}}

≤ C ∑_{n=1}^{∞} n^{β} ∑_{k=1}^{∞} P {|X| ≥ b_{nk}} ≤ C ∑_{n=1}^{∞} n^{β} ∑_{j=1}^{∞}(#I_{nj})P {|X| ≥ (nj)^{γ}}

≤ C ∑_{n=1}^{∞} n^{β} ∑_{j=1}^{∞}(#I_{nj})P {|X|^{1/γ} ≥ nj}

≤ C ∑_{n=1}^{∞} n^{β} ∑_{j=1}^{∞}(#I_{nj}) ∑_{k=nj}^{∞} P {k ≤ |X|^{1/γ} < k + 1}

= C ∑_{n=1}^{∞} n^{β} ∑_{k=n}^{∞} P {k ≤ |X|^{1/γ} < k + 1} ∑_{j=1}^{[k/n]}(#I_{nj})

≤ C ∑_{n=1}^{∞} n^{β} ∑_{k=n}^{∞} P {k ≤ |X|^{1/γ} < k + 1}n^{α+γ}([k/n] + 1)^{γθ}

≤ C ∑_{n=1}^{∞} n^{β}n^{α+γθ}n^{−γθ} ∑_{k=n}^{∞} k^{γθ}P {k ≤ |X|^{1/γ} < k + 1}

≤ C ∑_{k=1}^{∞} k^{γθ}P {k ≤ |X|^{1/γ} < k + 1} ∑_{n=1}^{k} n^{α+β} (∗)

≤ C ∑_{k=1}^{∞} k^{α+β+γθ+1}P {k ≤ |X|^{1/γ} < k + 1} because α + β ≠ −1

≤ CE|X|^{θ+ \frac{θ+α+β}{γ}} < ∞ by (6).

To verify (2), note that for any J > \frac{β+1}{γ(δ−θ)−α}

∑_{n=1}^{∞} n^{β} (∑_{k=1}^{∞} E||a_{nk}V_{nk}||^{δ})^{J} = ∑_{n=1}^{∞} n^{β} (∑_{k=1}^{∞} |a_{nk}|^{δ}E||V_{nk}||^{δ})^{J}

≤ ∑_{n=1}^{∞} n^{β} (sup_{k≥1} |a_{nk}|^{δ−θ} ∑_{k=1}^{∞} |a_{nk}|^{δ−θ}E||V_{nk}||^{δ})^{J}

≤ C ∑_{n=1}^{∞} n^{β} (n^{−γ(δ−θ)}n^{α}E|X|^{δ})^{J} (by (7), (8), and stochastic domination) = C ∑_{n=1}^{∞} n^{β+J(α−γ(δ−θ))} < ∞.

Finally, to verify (3) if β < 0, note that for any λ > 0

∑_{k=1}^{∞} P {||a_{nk}V_{nk}|| > λ} ≤ ∑_{k=1}^{∞} λ^{−δ}E||a_{nk}V_{nk}||^{δ} (by the Markov inequality)

≤ C sup_{k≥1} |a_{nk}|^{δ−θ} ∑_{k=1}^{∞} |a_{nk}|^{θ}E||V_{nk}||^{δ}

≤ Cn^{−γ(δ−θ)+α}E|X|^{δ} = o(1) (by (7), (8), and stochastic domination).

For the special case α + β = −1 we can establish the following result. Theorem 3.2. Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} is stochastically dominated by a random variable X. Assume that

sup_{k≥1} |a_{nk}| = O(n^{−γ}) for some γ > 0,

and

∑_{k=1}^{∞} |a_{nk}|^{θ} = O(n^{α}) for some 0 < θ ≤ 2 and any α, such that θ + \frac{α}{γ} < 2.

Let β = −1 − α and fix δ > θ such that \frac{α}{γ} + θ < δ ≤ 2. If E|X|^{δ} < ∞ and

S_{n} →^{P} 0,

then

∑_{n=1}^{∞} n^{β}P {||S_{n}|| > ε} < ∞ for all ε > 0.

Proof. Let c_{n} = n^{β}, n ≥ 1. Then we only need to verify that the conditions (1), (2), and (3) (if β < 0) of Theorem 1.1 hold with a_{nk}V_{nk} playing the role of V_{nk} in the formulation of that theorem. We mention that conditions (2) and (3) can be checked exactly in the same way as in the theorem above. While for condition (1) we need to make the following changes right after the inequality (*):

∑_{n=1}^{∞} n^{β} ∑_{k=1}^{∞} P {∥a_{nk}V_{nk}∥ ≥ 1}

≤ C ∑_{k=1}^{∞} k^{γθ}P {k ≤ |X|^{1/γ} < k + 1} ∑_{n=1}^{k} n^{α+β}

≤ C ∑_{k=1}^{∞} k^{γθ} log kP {k ≤ |X|^{1/γ} < k + 1} because α + β = −1

≤ CE|X|^{θ} log |X| ≤ CE|X|^{δ} < ∞ and since δ > θ.

Remarks. (i) We can verify that if θ ≤ 1, then it follows from the assumptions of Theorems 3.1 and 3.2 that the series S_{n} converges a.s. Note at the outset that the stochastic domination hypothesis ensures that E||V_{nk}|| ≤ CE|X|, k ≥ 1, n ≥ 1 and hence for all n ≥ 1, by the Beppo Levi theorem and the assumptions of Theorem 3.1

E (∑_{k=1}^{∞} ||a_{nk}V_{nk}||) = ∑_{k=1}^{∞} E||a_{nk}V_{nk}|| ≤ CE|X| ∑_{k=1}^{∞} |a_{nk}|

≤ C sup_{k≥1} |a_{nk}|^{1−θ} ∑_{k=1}^{∞} |a_{nk}|^{θ} ≤ Cn^{α−γ(1−θ)} < ∞.

Thus for all n ≥ 1, ∑_{k=1}^{∞} ||a_{nk}V_{nk}|| < ∞ a.s. and so for all n ≥ 1 and all K ≥ 1,

sup_{L>K} || ∑_{k=1}^{L} a_{nk}V_{nk} − ∑_{k=1}^{K} a_{nk}V_{nk}|| = sup_{L>K} || ∑_{k=K+1}^{L} a_{nk}V_{nk}||

≤ sup_{L>K} ∑_{k=K+1}^{L} ||a_{nk}V_{nk}|| = ∑_{k=K+1}^{∞} ||a_{nk}V_{nk}|| ⟶^{K→∞} 0 a.s.

Thus for all n ≥ 1, with probability 1, {∑_{k=1}^{K} a_{nk}V_{nk}, K ≥ 1} is a Cauchy sequence in X whence ∑_{k=1}^{∞} a_{nk}V_{nk} converges a.s. (ii) Take θ = 1 in order to obtain Theorem 1.3. (iii) Since Theorem 3.1 is stronger than Theorem 1.3, which is, in its turn is stronger than Theorem 1.2, the results of the papers Hsu and Robbins (1947), Hu, Moricz, and Taylor (1989), Gut (1992), Wang, Rao, and Yang (1993), Kuczmaszewska and Szynal (1994), Pruitt (1966), Rohatgi (1971), and Sung (1997) follow from it in the same way as was proved in Hu, Li, Rosalsky, and Volodin (2002).

(iv) Of course, the conditions of Theorems 3.1 and 3.2 can be slightly generalized on the case sup_{k≥1} |a_{nk}| = f(n) and ∑_{k=1}^{∞} |a_{nk}|^{θ} = g(n) where f(n) and g(n) are regularly varying functions with indexes −γ and α, correspondenly. The same is true for Theorems 3.3 and 3.4 below.

We can drop the convergence in probability condition if we assume that underling Banach space is of stable type.

Theorem 3.3. Assume that underlying Banach space has stable type p, 1 < p < 2. Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} has mean zero and is stochastically dominated by a random variable X. Moreover, assume that

sup_{k≥1} |a_{nk}| = O(n^{−γ}) for some γ > 0,

and

∑_{k=1}^{∞} |a_{nk}|^{θ} = O(n^{α}) for some 0 < θ ≤ p and any α such that θ + \frac{α}{γ} ≤ p.

Let β be such that α + β ≠ −1 and fix δ > θ such that θ + \frac{α}{γ} < δ ≤ 2. If

E|X|^{ν} < ∞, where ν = max(θ + \frac{1 + α + β}{γ} , δ, p)

then

∑_{n=1}^{∞} n^{β}P {||S_{n}|| > ε} < ∞ for all ε > 0.

Proof. We need to check that

S_{n} ≡ ∑_{k=1}^{∞} a_{nk}V_{nk} →^{P} 0.

To do this we apply Theorem 1.4. Since p ≥ θ + \frac{α}{γ}, we have

sup_{n≥1} ∑_{k=1}^{∞} |a_{nk}|^{p} = sup_{n≥1} ∑_{k=1}^{∞} |a_{nk}|^{p−θ}|a_{nk}|^{θ}

= sup_{n≥1}(sup_{k≥1} |a_{nk}|)^{p−θ} ∑_{k=1}^{∞} |a_{nk}|^{θ}
≤ C sup_{n≥1} n^{α−γ(p−θ)} < ∞.

All other assumptions of Theorem 3.1 are obviously satisfied.

32 A. VOLODIN, R. GIULIANO ANTONINI, AND T.-C. HU

For the special case α + β = −1 we can establish the following result. The proof repeats the proof of Theorem 3.3 with the only difference being, that we need to refer to Theorem 3.2 instead of Theorem 3.1. Because of this we omit the proof. Theorem 3.4. Assume that underlying Banach space has stable type p, 1 < p < 2. Suppose that the array {V_{nk}, k ≥ 1, n ≥ 1} has mean zero and is stochastically dominated by a random variable X. Moreover, assume that

sup_{k≥1} |a_{nk}| = O(n^{−γ}) for some γ > 0,

and

∑_{k=1}^{∞} |a_{nk}|^{θ} = O(n^{α}) for some 0 < θ ≤ p and any α such that θ + \frac{α}{γ} ≤ p.

Let β = −1−α and fix δ > θ such that θ+ \frac{α}{γ} < δ ≤ 2. If E|X|^{max(p,δ)} < ∞ then

∑_{n=1}^{∞} n^{β}P {||S_{n}|| > ε} < ∞ for all ε > 0.

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Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2. E-mail: andrei@math.uregina.ca;

Dipartimento di Matematica, Universitá di Pisa, 56100 Pisa, Italy. E-mail: giuliano@dm.unipi.it;

Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan. E-mail: tchu@math.nthu.edu.tw.