We consider a measurable stationary Gaussian stochastic process. A criterion for testing hypotheses about the covariance function of such a process using estimates for its norm in the space Lp(T), p≥1, is constructed.

Square Gaussian stochastic processcriterion for testing hypothesescorrelogram60G1062M07Introduction

We construct a criterion for testing the hypothesis that the covariance function of measurable real-valued stationary Gaussian stochastic process X(t) equals ρ(τ). We shall use the correlogram
ρˆ(τ)=1T∫0TX(t+τ)X(t)dt,0≤τ≤T,
as an estimator of the function ρ(τ).

A lot of papers so far have been dedicated to estimation of covariance function with given accuracy in the uniform metric, in particular, the papers [2, 4, 6, 11, 12] and the book [13]. We also note that similar estimates of Gaussian stochastic processes were obtained in books [7] and [1]. The main properties of the correlograms of stationary Gaussian stochastic processes were studied by Buldygin and Kozachenko [3].

The definition of a square Gaussian random vector was introduced by Kozachenko and Moklyachuk [10]. Applications of the theory of square Gaussian random variables and stochastic processes in mathematical statistics were considered in the paper [9] and in the book [3]. In the papers [5] and [8], Kozachenko and Fedoryanich constructed a criterion for testing hypotheses about the covariance function of a Gaussian stationary process with given accuracy and reliability in L2(T).

Our goal is to estimate the covariance function ρ(τ) of a Gaussian stochastic process with given accuracy and reliability in Lp(T), p≥1. Also, we obtain the estimate for the norm of square Gaussian stochastic processes in the space Lp(T). We use this estimate for constructing a criterion for testing hypotheses about the covariance function of a Gaussian stochastic process.

The article is organized as follows. In Section 2, we give necessary information about the square Gaussian random variables. In Section 3, we obtain an estimate for the norm of square Gaussian stochastic processes in the space Lp(T). In Section 4, we propose a criterion for testing a hypothesis about the covariance function of a stationary Gaussian stochastic process based on the estimate obtained in Section 3.

Some information about the square Gaussian random variables and processes([<xref ref-type="bibr" rid="j_vmsta17_ref_003">3</xref>]).

Let T be a parametric set, and let Ξ={ξt:t∈T} be a family of Gaussian random variables such that Eξt=0. The space SGΞ(Ω) is called a space of square Gaussian random variables if any ζ∈SGΞ(Ω) can be represented as
ζ=ξ¯TAξ¯−Eξ¯TAξ¯,
where ξ¯=(ξ1,…,ξN)T with ξk∈Ξ, k=1,…,n, and A is an arbitrary matrix with real-valued entries, or if ζ∈SGΞ(Ω) has the representation
ζ=limn→∞(ξ¯nTAξ¯n−Eξ¯nTAξ¯n).

A stochastic process Y is called a square Gaussian stochastic process if for each t∈T, the random variable Y(t) belongs to the space SGΞ(Ω).

An estimate for the <inline-formula id="j_vmsta17_ineq_027"><alternatives>
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<tex-math><![CDATA[$L_{p}(\mathbb{T})$]]></tex-math></alternatives></inline-formula> norm of a square Gaussian stochastic process

In the following theorem, we obtain an estimate for the norm of square Gaussian stochastic processes in the space Lp(T). We shall use this result for constructing a criterion for testing hypotheses about the covariance function of a Gaussian stochastic process.

Let{T,A,μ}be a measurable space, whereTis a parametric set, and letY={Y(t),t∈T}be a square Gaussian stochastic process. Suppose that Y is a measurable process. Further, let the Lebesgue integral∫T(EY2(t))p2dμ(t)be well defined forp≥1. Then the integral∫T(Y(t))pdμ(t)exists with probability 1, andP{∫T|Y(t)|pdμ(t)>ε}≤21+ε1/p2Cp1pexp{−ε1p2Cp1p}\varepsilon \bigg\}\le 2\sqrt{1+\frac{{\varepsilon }^{1/p}\sqrt{2}}{{C_{p}^{\frac{1}{p}}}}}\exp \bigg\{-\frac{{\varepsilon }^{\frac{1}{p}}}{\sqrt{2}{C_{p}^{\frac{1}{p}}}}\bigg\}\]]]>for allε≥(p2+(p2+1)p)pCp, whereCp=∫T(EY2(t))p2dμ(t).

Since maxx>0xαe−x=ααe−α0}{x}^{\alpha }{e}^{-x}={\alpha }^{\alpha }{e}^{-\alpha }$]]>, we have xαe−x≤ααe−α.

If ζ is a random variable from the space SGΞ(Ω) and x=s2·|ζ|Eζ2, where s>00$]]>, then
E(s2|ζ|Eζ2)α≤ααe−α·Eexp{s2|ζ|Eζ2}
and
E|ζ|α≤(2Eζ2s)αααe−αEexp{s2|ζ|Eζ2}.
From inequality (1) for 0<s<1 we get that
E|ζ|α≤(2Eζ2s)αααe−α(Eexp{s2ζEζ2}+Eexp{−s2ζEζ2})≤21−s(2Eζ2s)αααe−αexp{−s2}=2L0(s)(2Eζ2s)αααe−α.

Let Y(t), t∈T, be a measurable square Gaussian stochastic process. Using the Chebyshev inequality, we derive that, for all l≥1,
P{∫T|Y(t)|pdμ(t)>ε}≤E(∫T|Y(t)|pdμ(t))lεl.\varepsilon \bigg\}\le \frac{\mathbf{E}{(\int _{\mathbb{T}}|Y(t){|}^{p}d\mu (t))}^{l}}{{\varepsilon }^{l}}.\]]]>
Then from the generalized Minkowski inequality together with inequality (3) for l>11$]]> we obtain that
(E(∫T|Y(t)|pdμ(t))l)1l≤∫T(E|Y(t)|pl)1ldμ(t)≤∫T(2L0(s)(2EY2(t))pl2(pl)pls−plexp{−pl})1ldμ(t)=(2L0(s))1l∫T(2EY2(t))p2s−p(pl)pexp{−p}dμ(t)=(2L0(s))1l2p2s−p(pl)pexp{−p}∫T(EY2(t))p2dμ(t).

Assuming that Cp=∫T(EY2(t))p2dμ(t), we deduce that
E(∫T|Y(t)|pdμ(t))l≤2L0(s)2pl2(lp)plexp{−pl}Cpls−pl.
Hence,
P{∫T|Y(t)|pdμ(t)>ε}≤2·(2p2)lL0(s)(pp)l(exp{−p})lCpl(s−p)l·(lp)lεl=2L0(s)al(lp)l,\varepsilon \bigg\}& \displaystyle \le 2\cdot {\big({2}^{\frac{p}{2}}\big)}^{l}L_{0}(s){\big({p}^{p}\big)}^{l}{\big(\exp \{-p\}\big)}^{l}{C_{p}^{l}}{\big({s}^{-p}\big)}^{l}\cdot \frac{{({l}^{p})}^{l}}{{\varepsilon }^{l}}\\{} & \displaystyle =2L_{0}(s){a}^{l}{\big({l}^{p}\big)}^{l},\end{array}\]]]>
where a=2p2ppCpepspε, that is, a1p=212pCp1pesε1p. Let us find the minimum of the function ψ(l)=al(lp)l. We can easily check that it reaches its minimum at the point l∗=1ea1p.

Then
2L0(s)ψ(l∗)=2L0(s)a1ea1p·(1ea1p)p·1ea1p=2L0(s)a1ea1p·a−1ea1p·e−pea1p=2L0(s)exp{−pesε1p212peCp1p}=2L0(s)exp{−sε1p212Cp1p}=21−sexp{−s(12+ε1/p212Cp1p)}.

In turn, minimizing the function θ(s)=21−sexp{−s(12+ε1/p212Cp1p)} in s, we deduce s∗=1−11+2ε1/pCp1/p. Thus,
θ(s∗)=21+ε1/p2Cp1pexp{−ε1p2Cp1p}.
Since l∗≥1, it follows that inequality (2) holds if 1ea1p=sε1/p2pCp1/p≥1. Substituting the value of s∗ into this expression, we obtain the inequality ε2/p≥pCp1/p(Cp1/p+2ε1/p). Solving this inequality with respect to ε>00$]]>, we deduce that inequality (2) holds for ε≥(p2+(p2+1)p)pCp. The theorem is proved. □

The construction of a criterion for testing hypotheses about the covariance function of a stationary Gaussian stochastic process

Consider a measurable stationary Gaussian stochastic process X defined for all t∈R. Without any loss of generality, we can assume that X={X(t), t∈T=[0,T+A], 0<T<∞, 0<A<∞} and EX(t)=0. The covariance function ρ(τ)=EX(t+τ)X(t) of this process is defined for any τ∈R and is an even function. Let ρ(τ) be continuous on T.

Let the correlogramρˆ(τ)=1T∫0TX(t+τ)X(t)dt,0≤τ≤A,be an estimator of the covariance functionρ(τ). Then the following inequality holds for allε≥(p2+(p2+1)p)pCp:
P{∫0A(ρˆ(τ)−ρ(τ))pdτ>ε}≤21+ε1/p2Cp1pexp{−ε1p2Cp1p},\varepsilon \bigg\}\le 2\sqrt{1+\frac{{\varepsilon }^{1/p}\sqrt{2}}{{C_{p}^{\frac{1}{p}}}}}\exp \bigg\{-\frac{{\varepsilon }^{\frac{1}{p}}}{\sqrt{2}{C_{p}^{\frac{1}{p}}}}\bigg\},\]]]>whereCp=∫0A(2T2∫0T(T−u)(ρ2(u)+ρ(u+τ)ρ(u−τ))du)p2dτand0<A<∞.

Since the sample paths of the process X(t) are continuous with probability one on the set T, ρˆ(τ) is a Riemann integral.

Consider
E(ρˆ(τ)−ρ(τ))2=E(ρˆ(τ))2−ρ2(τ).
From the Isserlis equality for jointly Gaussian random variables it follows that
E(ρˆ(τ))2−ρ2(τ)=E(1T2∫0T∫0TX(t+τ)X(t)X(s+τ)X(s)dtds)−ρ2(τ)=1T2∫0T∫0T(EX(t+τ)X(t)EX(s+τ)X(s)+EX(t+τ)X(s+τ)×EX(t)X(s)+EX(t+τ)X(s)EX(s+τ)X(t))dtds−ρ2(τ)=1T2∫0T∫0T(ρ2(τ)+ρ2(t−s)+ρ(t−s+τ)ρ(t−s−τ))dtds−ρ2(τ)=1T2∫0T∫0T(ρ2(t−s)+ρ(t−s+τ)ρ(t−s−τ))dtds=2T2∫0T(T−u)(ρ2(u)+ρ(u+τ)ρ(u−τ))du.
We have obtained that
E(ρˆ(τ)−ρ(τ))2=2T2∫0T(T−u)(ρ2(u)+ρ(u+τ)ρ(u−τ))du.
Since ρˆ(τ)−ρ(τ) is a square Gaussian stochastic process (see Lemma 3.1, Chapter 6 in [3]), it follows from Theorem 2 that
P{∫0A(ρˆ(τ)−ρ(τ))pdτ>ε}≤21+ε1/p2Cp1pexp{−ε1p2Cp1p}.\varepsilon \bigg\}\le 2\sqrt{1+\frac{{\varepsilon }^{1/p}\sqrt{2}}{{C_{p}^{\frac{1}{p}}}}}\exp \bigg\{-\frac{{\varepsilon }^{\frac{1}{p}}}{\sqrt{2}{C_{p}^{\frac{1}{p}}}}\bigg\}.\]]]>
Applying Eq. (5), we get
Cp=∫0A(2T2∫0T(T−u)(ρ2(u)+ρ(u+τ)ρ(u−τ))du)p2dτ.
The theorem is proved. □

Denote
g(ε)=21+ε1/p2Cp1pexp{−ε1p2Cp1p}.
From Theorem 3 it follows that if ε≥zp=Cp(p2+(p2+1)p)p, then
P{∫0A(ρˆ(τ)−ρ(τ))pdτ>ε}≤g(ε).\varepsilon \bigg\}\le g(\varepsilon ).\]]]>
Let εδ be a solution of the equation g(ε)=δ, 0<δ<1. Put Sδ=max{εδ,zp}. It is obvious that g(Sδ)≤δ and
P{∫0A(ρˆ(τ)−ρ(τ))pdτ>Sδ}≤δ.S_{\delta }\bigg\}\le \delta .\]]]>

Let H be the hypothesis that the covariance function of a measurable real-valued stationary Gaussian stochastic process X(t) equals ρ(τ) for 0≤τ≤A. From Theorem 3 and (6) it follows that to test the hypothesis H, we can use the following criterion.

For a given level of confidence δ the hypothesisHis accepted if∫0A(ρˆ(τ)−ρ(τ))pdμ(τ)<Sδ;otherwise, the hypothesis is rejected.

The equation g(ε)=δ has a solution for any δ>00$]]> since g(ε) is a decreasing function. We can find the solution of the equation using numerical methods.

We can easily see that Criterion 1 can be used if Cp→0 as T→∞.

The next theorem contain assumptions under which Cp→0 as T→∞.

Letρ(τ)be the covariance function of a centered stationary random process. Letρ(τ)be a continuous function. Ifρ(T)→0asT→∞, thenCp→0asT→∞, whereCp=∫0A(ψ(T,τ))p/2dtandψ(T,τ)=2T2∫0T(T−u)(ρ2(u)+ρ(u+τ)ρ(u−τ))du,A>0,T>0.0,\hspace{2.5pt}T>0.\]]]>

We have ψ(T,τ)≤2T∫0T(ρ2(u)+ρ(u+τ)ρ(u−τ))du≤4ρ2(0). Now it is suffices to prove that ψ(T,τ)→0 as T→∞. From the L’Hopital’s rule it follows that
limT→∞ψ(T,τ)=limT→∞2T∫0T(ρ2(u)+ρ(u+τ)ρ(u−τ))du=limT→∞(ρ2(T)+ρ(T+τ)ρ(T−τ))=0.

Application of Lebesgue’s dominated convergence theorem completes the proof. □

Here are examples in which we find the estimates for Cp.

Let H be the hypothesis that the covariance function of a centered measurable stationary Gaussian stochastic process equals ρ(τ)=Bexp{−a|τ|}, where B>00$]]> and a>00$]]>.

To test the hypothesis H, we can use Criterion 1 by selecting ρˆT(τ) that is defined in (4) as an estimator of the function ρ(τ). Let 0<A<∞. We shall find the value of the expression
I=∫0T(T−u)(e−2au+e−a|u+τ|e−a|u−τ|)du=∫0TTe−2audu+T∫0Te−a|u+τ|e−a|u−τ|du−∫0Tue−2audu−∫0Tue−a|u+τ|e−a|u−τ|du=I1+I2+I3+I4.
Now lets us calculate the summands:
I1=T∫0Te−2audu=T2a(1−e−2aT),I2=T∫0Te−a|u+τ|e−a|u−τ|du=T(∫0τe−a(u+τ)ea(u−τ)du+∫τTe−a(u+τ)e−a(u−τ)du)=T(∫0τe−2aτdu+∫τTe−2audu)=T(τe−2aτ−12ae−2aT+12ae−2aτ),I3=∫0Tue−2audu=−T2ae−2aT+12a∫0Te−2audu=−T2ae−2aT−14a2e−2aT+14a2,I4=∫0Tue−a|u+τ|e−a|u−τ|du=∫0τue−a(u+τ)ea(u−τ)du+∫τTue−a(u+τ)e−a(u−τ)du=∫0τue−2aτdu+∫τTue−2audu=τ22e−2aτ−T2ae−2aT+τ2ae−2aτ−14a2e−2aT+14a2e−2aτ.
Therefore,
I=(Tτ+T2a−τ22−τ2a−14a2)e−2aτ+12a2e−2aT+T2a−14a2≤(Tτ+T2a)e−2aτ+T2a+12a2e−2aT.

Thus, we obtain
Cp≤(2BT2)p2∫0A((Tτ+T2a)e−2aτ+T2a+12a2e−2aT)p/2dτ=(2B)p2Tp/2TpI5=(2B)p21Tp/2I5,
where I5=∫0A((τ+12a)e−2aτ+12a+12a2e−2aT)p/2dτ.

Let H be the hypothesis that the covariance function of a centered measurable stationary Gaussian stochastic process equals ρ(τ)=Bexp{−a|τ|2}, where B>00$]]> and a>00$]]>.

Similarly as in the previous example, to test the hypothesis H, we can use Criterion 1 by selecting ρˆT(τ) defined in (4) as the estimator of the function ρ(τ). Let 0<A<∞. Let us find the value of the expression
I=∫0T(T−u)(e−2au2+e−a|u+τ|2e−a|u−τ|2)du=∫0TTe−2au2du+T∫0Te−a|u+τ|2e−a|u−τ|2du−∫0Tue−2au2du−∫0Tue−a|u+τ|2e−a|u−τ|2du=I1+I2+I3+I4.
Now let us calculate the summands:
I1=T∫0Te−2au2du≤T∫0∞e−2au2du=πT22a,I2=T∫0Te−a|u+τ|2e−a|u−τ|2du=Te−2aτ2∫0Te−2au2du≤πT22ae−2aτ2,I3=∫0Tue−2au2du=−14a∫0Te−2au2d(−2au2)=−14a(e−2aT2−1),I4=∫0Tue−2a(u2+τ2)du=e−2aτ2∫0Tue−2au2du=−14ae−2aτ2(e−2aT2−1).
Hence,
I≤πT22a+πT22ae−2aτ2+14a(e−2aT2−1)+14ae−2aτ2(e−2aT2−1)≤T(π22a+π22ae−2aτ2).
Thus, we obtain
Cp≤(2BT2)p2∫0A(T(π22a+π22ae−2aτ2))p/2dτ=(2B)p21Tp/2I6,
where I6=∫0A(π22a+π22ae−2aτ2)p/2dτ.

Acknowledgments

The authors express their gratitude to the referee and Professor Yu. Mishura for valuable comments that helped to improve the paper.

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