����������� ������ 47 (1995), 31–37

UDC 519.246

���������� ������ ��� research paper

ON PROBABILITY MEASURE GENERATED BY SPECTRAL DENSITY ESTIMATE

Pavle Mladenović

Abstract. We study the periodogram based estimate of spectral density of a strictly stationary random sequence and consider this estimate as a random function on the whole domain of frequency. We renormalize the scale in this domain and investigate the probability measure generated by obtained process in the space of continuous functions.

1. Introduction

In this paper we shall consider a strictly stationary real random sequence X(t), t ∈ D = {…, −1, 0, 1,… }, with the mean EX(t) = 0 and the spectral density f(λ), λ ∈ [−π, π]. Let (X(1),…, X(N) be a sample of size N from this random sequence. The spectral density estimate of the Grenander-Rosenblatt type is given by

f̂_{N}(λ) = ∫_{−π}^{π} φ_{N}(x − λ)\frac{1}{2πN} |∑_{t=1}^{n} X(t)e^{−itx}|^{2} dx,

where: φ_{N}(x) = B_{N}^{−1}φ(xB_{N}^{−1}), x ∈ [−π, π]; φ : R → R is a weight function for which we suppose that is symetric about 0, has a bounded first derivative and such that φ(0) = 1, ∫_{−π}^{π} φ(x) dx = 1 and φ(x) = 0 for |x| ≥ π. The sequence (B_{N}) of real numbers is such that B_{N} → 0 and NB_{N} → ∞ when N → ∞. We assume that

B_{N} = N^{−ε}, where \frac{1}{3} < ε < \frac{1}{2}, and that the functions f and φ_{N} are defined on the whole real line R and 2π-periodic.

Let us define the random process ξ̃_{N}(λ) and Z̃_{N}(λ) as follows:

ξ̃_{N}(λ) = \sqrt{NB_{N}} [f̂_{N}(λB_{N}) − Ef̂_{N}(λB_{N})] , |λ| ≤ πB_{N}^{−1},
Z̃_{N}(λ) = \sqrt{NB_{N}} [f̂_{N}(λB_{N}) − f(λB_{N})] , |λ| ≤ πB_{N}^{−1},

AMS Subject Classification: 62M15, 60G10, 60G15.

Keywords: stationary time series, spectral density, periodogram based estimate, cumulant spectral densities, weak convergence.

and let Z(λ), λ ∈ R, be the Gaussian random process with the mean EZ(λ) = 0 and the covariance function

R(λ, μ) = EZ(λ)Z(μ) = 2πf^{2}(0) ∫_{−∞}^{∞} φ(x − λ)[φ(x − μ) + φ(x + μ)] dx.

The cumulant spectral density of order n of the sequence X(t), t ∈ D, is defined as follows:

f_{n}(λ_{1}, λ_{2},…, λ_{n−1}) = \frac{1}{(2π)^{n−1}} ∑_{t_{1},t_{2},...,t_{n−1}} S_{n}(t_{1},…, t_{n−1}, 0) exp(− ∑_{j=1}^{n−1} λ_{j}t_{j}),

where S_{n}(t_{1},…, t_{n−1}, 0) denote the n–th cumulant of the sequence X(t), t ∈ D. (The function f_{n} is defined if the series on the right–hand side of (1) converges.) Of course, the spectral density f is the cumulant spectral density of second order.

We shall also use the following denotations:

C = C[a, b]—the space of all real continuous funstions defined on [a, b], where −∞ < a < b < +∞, with the uniform metric ρ(x, y) = sup_{a≤t≤b} |x(t) − y(t)|;

C—the class of Borel sets in C;

Lip_{H}(1, ∞)—the class of real functions g which satisfy the condition

(∀x, y) |g(x) − g(y)| ≤ H|x − y|;

P_{N}, Q_{N} and P —the probability measures generated by the random processes ξ̃_{N}(λ), a ≤ λ ≤ b, Z̃_{N}(λ), a ≤ λ ≤ b, and Z(λ), a ≤ λ ≤ b, respectively, on the space (C, C).

2. Weak convergence of probability measure Q_{N}

The spectral density estimate of Grenander-Rosenblatt type have been considered in many papers. See, for example, [6] for references. A very important result is that the spectral density estimate has asymptoticaly normal distribution [3,6]. In the papers [4,5] the asymptotic properties of the random process ξ̃_{N}(λ) have been investigated. Specially, the weak convergence of the mesure P_{N} has been proved. Here we shall prove the following two theorems:

THEOREM 1. Let all cumulant spectral densities of the random process X(t), t ∈ D, are bounded and let the spectral density function f be continuously-differentiable. Then, the finite-dimensional distributions of the random process Z̃_{N}(λ), λ ∈ R, converge weakly to those of the Gaussian process Z(λ), λ ∈ R.

THEOREM 2. Let the sequence X(t) be Gaussian and f′ ∈ Lip_{H}(1, ∞). Then, Q_{N} converges weakly to P, when N → ∞.

On probability measure generated by spectral density estimate 33

3. Proofs

In order to prove these two theorems we need several lemmas:

LEMMA 1. Let the first derivative of the spectral density be bounded, i.e. ∥f′∥ = C < +∞. Then, the following inequality is valid:

sup_{−π≤λ≤π} |f(λ) − Ef̂_{N}(λ)| ≤ C ∫_{−π}^{π} |xφ_{N}(x)| dx + o(∫_{−π}^{π} |xφ_{N}(x)| dx).

Proof. Let us denote I_{N}(x) = \frac{1}{2πN} |∑_{t=1}^{N} X(t)e^{−itx}|^{2}. Then we have

Ef̂_{N}(λ) − f(λ) = E ∫_{−π}^{π} φ_{N}(x)I_{N}(x + λ) dx − f(λ)

= \frac{1}{2πN} ∫_{−π}^{π} φ_{N}(x) E(∑_{t=1}^{N} X(t)e^{−it(x+λ)} ∑_{s=1}^{N} X(s)e^{is(x+λ)}) dx − f(λ)

= \frac{1}{2πN} ∫_{−π}^{π} φ_{N}(x) ∑_{t=1}^{N} ∑_{s=1}^{N} EX(t)X(s)e^{i(x+λ)(s−t)} dx − f(λ).

Using the equalities EX(t) = 0 and EX(t)X(s) = ∫_{−π}^{π} e^{iλ(t−s)}f(λ) dλ we obtain

Ef̂_{N}(λ) − f(λ) = \frac{1}{2πN} ∫_{−π}^{π} φ_{N}(x) ∑_{t=1}^{N} ∑_{s=1}^{N} ∫_{−π}^{π} f(u)e^{iu(t−s)}e^{i(x+λ)(s−t)}du dx − f(λ)

= ∫_{−π}^{π} ∫_{−π}^{π} φ_{N}(x)f(u)\frac{1}{2πN} ∑_{t=1}^{N} ∑_{s=1}^{N} e^{i(u−x−λ)(t−s)}du dx − f(λ).

Using the equality ∑_{t=1}^{N} e^{itx} = \frac{sin(Nx/2)}{sin(x/2)}e^{i(N+1)x/2}, we get

\frac{1}{2πN} ∑_{t=1}^{N} ∑_{s=1}^{N} e^{ix(t−s)} = \frac{1}{2πN} \begin{matrix}sin & 2 & \frac{Nx}{2} \\ sin & 2 & \frac{x}{2} \end{matrix} = Φ_{N}(x),

where Φ_{N} is Fejér’s kernel and Ef̂_{N}(λ) − f(λ) = ∫_{−π}^{π} ∫_{−π}^{π} φ_{N}(x)Φ_{N}(x + λ − u)f(u) du dx − f(λ).

Using the periodicity of the functions φ_{N}, Φ_{N} and f and the fact that the functions φ_{N} and Φ_{N} are even, we get

Ef̂_{N}(λ) − f(λ) = ∫_{−π}^{π} ∫_{−π}^{π}(f(λ + u) − f(λ))φ_{N}(u)Φ_{N}(x) dx

+ ∫_{−π}^{π} ∫_{−π}^{π}(f(λ + u) − f(λ))(φ_{N}(x + u) − φ_{N}(u))Φ_{N}(x) du dx

≤ C ∫_{−π}^{π} |uφ_{N}(u)| du + o(∫_{−π}^{π} |xφ_{N}(x)| dx).

LEMMA 2. [4] Let X(t), t ∈ D, be a stationary random sequence with the mean EX(t) = 0, the spectral density f with ∥f′∥ = C < +∞ and the spectral density of order four for which we have sup |f_{4}(u_{1}, u_{2}, u_{3}, u_{4})| = C_{1} < +∞. For every λ, μ ∈ [−π, π] the covariance of the random variable f̂_{N}(λ) and f̂_{N}(μ) is given by

cov(f̂_{N}(λ), f̂_{N}(μ)) = \frac{2π}{N} f(λ)f(μ) ∫_{−π}^{π} φ_{N}(x − λ)[φ(x − μ) + φ_{N}(x + μ)]dx

+ |λ − μ|O (\frac{ln^{2} N}{N} ∫_{−π}^{π} φ_{N}^{2}(x)dx) + o (\frac{1}{N} ∫_{−π}^{π} φ_{N}^{2}(x)dx)

+ O (\frac{ln^{4} N}{N^{2}} ∫_{−π}^{π} x^{2}φ_{N}^{2}(x)dx ∫_{−π}^{π} φ_{N}^{2}(x)dx) ,

when N → ∞, uniformly for λ, μ ∈ [−π, π].

LEMMA 3. Let the conditions of Theorem 1 be satisfied. Then, for every λ and μ the following equalities are valid:

lim_{N→∞} cov(ξ̃_{N}(λ), ξ̃_{N}(μ)) = lim_{N→∞} cov(Z̃_{N}(λ), Z̃_{N}(μ)) = R(λ, μ).  (1)

Proof. The first of the equalities (1), as a consequence of Lemma 2, was proved

in [4]. Using the fact that B_{N} = N^{−ε}, where \frac{1}{3} < ε < \frac{1}{2}, and Lemma 1 we get

sup_{|λ|≤πB_{N}^{−1}} |ξ̃_{N}(λ) − Z̃_{N}(λ)| = sup_{−π≤λ≤π} \sqrt{NB_{N}}|Ef̂_{N}(λ) − f(λ)|

= O(\sqrt{NB_{N}} ∫_{−π}^{π} |xφ_{N}(x)| dx) = O(\sqrt{NB_{N}} ∫_{−πB_{N}^{−1}}^{πB_{N}^{−1}} B_{N}|tφ(t)| dt)

= O(N^{1/2}B_{N}^{3/2}) = o(1), N → ∞,

and the second of the equalities (1) follows.

LEMMA 4. [1] Let all cumulant spectral densities of the random sequence X(t), t ∈ D, be bounded. For the cumulants of the random process f̂_{N}(λ), −π ≤ λ ≤ π, the following inequality is valid:

|S_{N}(f̂_{N}(λ_{1}), f̂_{N}(λ_{2}),…, f̂_{N}(λ_{n}))| ≤ \frac{K_{n}}{(NB_{N})^{n−1}} .

The constant K_{n} does not depend on the particular choice of points λ_{1}, …, λ_{n}.

Proof of Theorem 1. It follows from Lemmas 1, 3 and 4 that all cumulants of the random process Z̃_{N}(λ) converge to those of the process Z(λ) and this completes the proof.

LEMMA 5. [5] Let the sequence X(t), t ∈ D, be Gaussian and its spectral density f bounded. Then, there exist constants K > 0, η > 0 and ε > 0 such that the inequality E|ξ̃_{N}(λ) − ξ̃_{N}(μ)|^{η} ≤ K|λ − μ|^{1+ε} holds for every N, λ and μ.

LEMMA 6. Let f : R → R be continuously-differentiable function and f′ ∈ Lip_{H}(1, ∞). Then, for every λ, μ ∈ R the following inequality is valid:

Δ_{f}(λ, μ, u) := |f(λ + u) − f(λ) − f(μ + u) + f(μ)| ≤ 2H|u| · |λ − μ|.

Proof. From the fact that the function f is differentiable it follows that the following equalities are valid:

f(λ + u) − f(λ) = uf′(λ + ϑ_{1}u), 0 ≤ ϑ_{1} ≤ 1,  (2)
f(μ + u) − f(μ) = uf′(μ + ϑ_{2}u), 0 ≤ ϑ_{2} ≤ 1,  (3)
f(λ) − f(μ) = (λ − μ)f′(μ + ϑ_{3}(λ − μ)), 0 ≤ ϑ_{3} ≤ 1,  (4)
f(λ + u) − f(μ + u) = (λ − μ)f′(μ + u + ϑ_{4}(λ − μ)), 0 ≤ ϑ_{4} ≤ 1.  (5)

Using the fact that f′ ∈ Lip_{H}(1, ∞) and the equalities (2)–(5) we get

Δ_{f}(λ, μ, u) = |u| · |f′(λ + ϑ_{1}u) − f′(μ + ϑ_{2}u)|
≤ H|u| · |λ − μ + (ϑ_{1} − ϑ_{2})u| ≤ H|u| · |λ − μ| + H|u|^{2},
Δ_{f}(λ, μ, u) = |λ − μ| · |f′(μ + ϑ_{3}(λ − μ) − f′(μ + u + ϑ_{4}(λ − μ))|

≤ H|λ − μ| · |u + (ϑ_{4} − ϑ_{3})(λ − μ)| ≤ H|u| · |λ − μ| + H|λ − μ|^{2}.

From these inequalities it follows that

Δ_{f}(λ, μ, u) ≤ H|u| · |λ − μ| + min{Hu^{2}, H|λ − μ|^{2}}
≤ H|u| · |λ − μ| + H|u| · |λ − μ| = 2H|u| · |λ − μ|.

LEMMA 7. [6] For every integrable function h, with |h(x)| ≤ H|x| and Fejér’s kernel Φ_{N} the following equality is valid:

∫_{−π}^{π} ∫_{−π}^{π} h(x)Φ_{N}(x)(φ_{N}(x + z) − φ_{N}(x)) dx dz = o(∫_{−π}^{π} |xφ_{N}(x)| dx).

DEFINITION. For the (random) function ξ(λ), λ ∈ Λ ⊂ R we define the modulus of continuity ω_{ξ}(δ) in the following way:

ω_{ξ}(δ) = sup_{|λ−μ|<δ, λ,μ∈Λ} |ξ(λ) − ξ(μ)|.

LEMMA 8. Let f′ ∈ Lip_{H}(1, ∞). Then, we have
ωZ̃_{N} (δ) ≤ ωξ̃_{N} (δ) + ω(δ),  (6)

where ω(δ) ↓ 0 when δ ↓ 0 and the function ω(·) does not depend on N.

36 P. Mladenović

Proof. Since

sup_{|λ−μ|<δ}|Z̃_{N}(λ) − Z̃_{N}(μ)| ≤ sup_{|λ−μ|<δ} |ξ̃_{N}(λ) − ξ̃_{N}(μ)|

+ sup_{|λ−μ|<δ} |Z̃_{N}(λ) − ξ̃_{N}(λ) − (Z̃_{N}(μ) − ξ̃_{N}(μ))|,

we obtain that ωZ̃_{N} (δ) ≤ ωξ̃_{N} (δ) + ω_{Z̃_{N} −ξ̃_{N}} (δ). It is sufficient to prove that ω_{Z̃_{N} −ξ̃_{N}} (δ) is bounded and tends to 0 when δ tends to 0.

ω_{Z̃_{N} −ξ̃_{N}} (δ) = sup_{|λ−μ|<δ} \sqrt{NB_{N}}|f(λ) − Ef̂_{N}(λ) − f(μ) + Ef̂_{N}(μ)|

= sup_{|λ−μ|<δ} \sqrt{NB_{N}} | ∫_{−π}^{π}(f(λ + u) − f(λ) − f(μ + u) + f(μ))φ_{N}(u) du

+ ∫_{−π}^{π} ∫_{−π}^{π}(f(λ + u) − f(λ) − f(μ + u) + f(μ))(φ_{N}(u + x) − φ_{N}(u))Φ_{N}(x) du dx|

≡ sup_{|λ−μ|<δ} \sqrt{NB_{N}}|A_{1} + A_{2}|.

If |λ − μ| ≤ δ then using Lemma 6 we get

\sqrt{NB_{N}}|A_{1}| ≤ 2H|λ − μ|\sqrt{NB_{N}} ∫_{−π}^{π} |uφ_{N}(u)| du
≤ 2HδN^{1/2}B_{N}^{3/2} ∫_{−∞}^{+∞} |xφ(x)| dx ≤ C′_{N}δ ↓ 0, δ ↓ 0,

Note that the sequence C′_{N} = 2HN^{1/2}B_{N}^{3/2} ∫_{−∞}^{+∞} |xφ(x)| dx, N = 1, 2, 3,… is bounded. Using Lemmas 6 and 7 and the inequality |λ−μ| ≤ δ we obtain \sqrt{NB_{N}}|A_{2}| ≤ C′′_{N}δ, where

C′′_{N} = \sqrt{NB_{N}}o(∫_{−π}^{π} |xφ_{N}(x)| dx) = o(1), N → ∞..

Hence, for ω(δ) = δ · sup_{N}(C′_{N} + C′′_{N}) the inequality (6) is valid.

Proof of Theorem 2. The sequence (Q_{N}) is tight if and only if the following two conditions hold [2]:

(a) For each η > 0, there exists a constant a such that P {|Z̃_{N}(0)| > a} ≤ η, for N ≥ 1;

(b) For each ε > 0 and η > 0, there exist a constant δ ∈ (0, 1) and an integer n_{0}, such that

P {ωZ̃_{N} (δ) ≥ ε} ≤ η, for N ≥ n_{0}.

The condition (a) follows from the fact that Z̃_{N}(λ) has asimptotically normal distribution, and the conditions (b) follows from Lemmas 5 and 8. Since the sequence (Q_{N}) is tight and Theorems 1 holds, it follows that Q_{N} converge weakly to P.

REFERENCES

[1] Bentkus, R., On cumulants of the spectrum estimate of a stationary time series, Lietuvos matematikos rinkinys, 16, 4 (1976), 37–61.

[2] Bilingsley, P., Convergence of Probability Measures, Wiley, New York, 1968.

[3] Brillinger, D. B., Asymptotic properties of spectral estimates of second order, Biometrika, 56 (1969), 375–390.

[4] Mladenović, P., On covariance of spectral estimates of stationary random sequence, Publ. Inst. Math. 49 (63) (1991), 239–245.

[5] Mladenović, P., On weak convergence of spectral density estimate, Publ. Inst. Math. 49 (63) (1991), 233–238.

[6] Žurbenko, I., The Spectral Analysis of Time Series, North–Holland, 1986.

(received 23.06.1995.)

Mathematical faculty Belgrade University Studentski trg 16, 11000 Belgrade YUGOSLAVIA