Proper complex random processes

In this paper we study properties of stationary proper complex random process with stable correlation functions. Estimates are obtained for distribution of supremum of modulus of these processes and normes in spaces Lp on finite and infinite intervals.


Introduction
This article deals with complex random processes which are one of the most important generalizations of the concept of random process (see [2,12]).The complex random processes are especially relevant when the narrowbanded processes are investigated.These processes are exploited as models of complex amplitudes of quasiharmonic oscillations or waves in radiophysics and optics [1].In this article we presented results of investigation of properties of complex random processes which are useful when solving problems in the listed above areas.Conditions for existence of proper complex random processes are described in [12,2].In this article we investigate stationary proper complex random processes, stationary proper complex random processes with stable correlation function.Some results for properties of stable correlation function are presented in paper [11].In this article some properties of square Gaussian random variables and process are presented (for more results see, for example, [3,7,8]).Also, in this paper estimates of distributions of functionals from the module of stationary Gaussian proper complex random processes are obtained (for more results see, for example, [13,6,10]).Theorems, which describe behavior of the module of stationary proper complex random process at infinity are developed.
The content of the article is as follows.In Section 2 we introduce the basic definitions related to the complex random processes.Stationary proper complex random processes are introduced and discussed in Section 3. In the next Section 4, we deal with properties of square Gaussian random variables and processes.Section 5 is related to estimates of distributions of some functions from the module of stationary Gaussian proper complex random process.And in the last Section 6 behavior of the module of stationary proper complex random process at infinity is studied.

Proper Complex Random Process
Definition 2.1.A random process of the form X(t) = X c (t) + iX s (t), t ∈ R, where X c (t) and X s (t) are realvalued random processes (c -cosine, s -sine), is called complex random process (see book [2] and paper [12]).
Remark 2.1.In this paper we will consider centered random processes, that is is called correlation function of the process X(t).
The function is called pseudo correlation function of the process X(t).
Definition 2.3.A complex random process X(t) is called proper complex random process (PCR process), if the pseudo correlation function of this process is equal to zero, EX(t + τ )X(t) = 0, that is when conditions hold true.
Remark 2.2.Conditions under which PCR processes exist are described in book [2] and paper [12].
Remark 2.3.In the case where PCR-process X(t) is stationary we can write the following relations Definition 2.5.A complex random process X(t) = X c (t) + iX s (t) is called Gaussian if the real-valued random processes X c (t) and X s (t) are jointly Gaussian processes.

Stationary PCR-processes with stable correlation functions
Definition 3.1.The correlation function r(τ ), τ ∈ R of stationary proper complex random process is called stable correlation function if it can be represented in the form where σ 2 , c, β, α are real-valued constants, such that Stat., Optim.Inf.Comput.Vol.
where the function r(τ ) is given by formula (5).
Remark 3.2.For the proper stationary random complex process X(t) = X c (t) + iX s (t), EX(t) = 0, with stable covariance functions the following relations hold true

Square Gaussian random variables and processes
In this section we propose definitions and some properties of square Gaussian random variables and processes.
Definition 4.1.[3, 7] Let (T, ρ) be a metric space and let Θ = {ξ(t), t ∈ T }, Eξ(t) = 0, be a family of jointly Gaussian random variables (e.g.ξ = {ξ(t), t ∈ T } is a Gaussian random process).The space of square Gaussian random variable (SG Θ (Ω)) is such a space that any element η ∈ SG Θ (Ω) can be presented in the form where and A is a real-valued matrix, or the element η ∈ SG Θ (Ω) is the mean square limit of a sequence of random variables of the form (9): ) .
Definition 4.2.A random process η = {η(t), t ∈ T } is called square Gaussian process if the family of random variables η = {η(t), t ∈ T } forms the space of square Gaussian random variables.
The next theorem is a modification of Theorem 3.2 from the book [7].
]} be a separable square Gaussian random process and let the condition holds true for β ∈ (0, 1], C > 0. Then for all integer M > 1 and all where γ 0 = sup a≤t≤b (V arX(t))  where the following inequality holds true 5. Estimation of distribution of some functionals from module of stationary Gaussian PCR-processes Theorem 5.1

Proof
The proof of this theorem follows from inequality (13).Indeed it follows from (6), that Therefore Suppose that (X 1 , X 2 , X 3 , X 4 ) is a zero-mean Gaussian vector.Then we have: This equality is called Isserlis formula (see, for example [3, p.228].Making use of this formula and relations (3), ( 4) we can write It follows from (15) that

Now (14) follows from (13).
Theorem 5.2 Let X = {X(t), t ∈ [a, b]} be a Gaussian SPCR process and let If X(t) is a separable process, then for all integer M > 1 and all

Proof
The statement of this Theorem follows from Theorem 4.1.In our case γ 0 = σ 2 .
In order to to apply Theorem 4.1 to the process ) 2 we have to astimate Next¡ we have Since Im (r(0)) = 0, then In the same way we can obtaine that Consequently we have we get the following estimate Consequently β = α 2 and C = 2σ 2 √ c.Therefore (16) follows from (11).

Proof
The statement of this Theorem follows from Theorem 4.2 (see Corollary 4.1) if we take Corollary 6.1 Let c(t) > 0 be an even monotone increasing function for which conditions of Theorem 6.1 are satisfied.Then for all t ≥ 0 the following inequality holds true with probability one: The function r(τ ) is non-negative definite, since r(τ ) is the characteristic function of a stable random variable ξ, Eξ = 0, in the case where σ 2 = 1 (see[11, p.169]).