Ergodicity is a fundamental property in statistical information signal modelling and processing because it simplifies the analysis and estimation of signal properties and system parameters. It enables practitioners to work with single realizations of signals and make meaningful statistical inferences, which is necessary when dealing with real-world data and signals. The continuoustime stationary conditional linear random process as a mathematical model of information signals has been analyzed in the paper using characteristic functions method. The mixing property of the process, from which its ergodicity follows, has been proven.
Conditional linear random process (CLRP) is represented as a stochastic integral with a random
kernel driven by the process with independent increments [
Ergodicity is always the important property of mathematical models which is used for information
signal processing when the task is to estimate parameters of a signal or a system [
Very often the ergodic property of the investigated signal is just a hypothesis or assumption.
However it is a characteristic property of linear random processes [
The main goal of the paper is to justify the conditions for the continuous-time stationary conditional linear random process to be ergodic as a consequence of mixing property using the known representation of its multidimensional characteristic function.
In the following parts of the article, we define the continuous-time conditional linear random process and represent its multidimensional characteristic function. Then we consider the notions of ergodicity and mixing in terms of characteristic functions of the stationary random processes. Finally, we find the conditions of stationary CLRP to be mixing and ergodic.
This section is preliminary and covers the definition of continuous-time conditional linear random
process and representation of its multidimensional characteristic function which is used in the next
section for proving the mixing property and ergodicity. Here we follow mostly the results of [
A continuous-time conditional linear random process ξ(ω,t), ω∈ Ω, t ∈ (−∞, ∞) (where {Ω, F, P} is probability space) is defined as the following stochastic integral:
∞ ξ(ω,t) = ∫ ϕ(ω, τ,t)dη(ω, τ), ω∈ Ω, t ∈ ,
−∞ where ϕ(ω, τ,t), τ,t ∈ is a stochastic kernel; η(ω, τ), τ ∈ (−∞, ∞) is a Levy process; random functions ϕ(ω, τ,t) and η(ω, τ) are stochastically independent.
m ∞ fξFϕ (ω,u1,u2 , ..., um ; t1,t2 , ..., tm ) =ia exp ∑ uk ∫ ϕ(ω, τ,tk )d τ +
k=1 −∞ ∞ ∞ m + ∫ ∫ exp ix ∑uk ϕ(ω, τ,tk ) −1 − −∞ −∞ k=1 ix ( k∑=1uk ϕ(ω, τ,tk )) 1 + x2 m 1 + x2 x2
dG(x)d τ ,
uk , tk ∈ (−∞, ∞), k = 1, m , where G(x), x ∈ is a real non-decreasing and bounded function such that G(−∞) = 0 (the Poisson jump spectrum in Levy-Khintchine form of infinitely divisible Levy process η(ω, τ) ); Let Fϕ ⊂ F be a σ -subalgebra generated by the random kernel ϕ(ω, τ,t) satisfying the condition (1) (2) ∞ ∫ ϕ(ω, τ,t) < ∞ with probability 1. −∞
The m-dimensional characteristic function of conditional linear random process (1) is represented in the following form:
m fξ (u1,u2 ,...,um ;t1,t2 ,...,tm ) =E exp i∑ uk ξ(ω,tk ) =EfξFϕ (ω,u1,u2 ,...,um ;t1,t2 ,...,tm ), k=1 m where fξFϕ (ω,u1,u2 ,...,um ;t1,t2 ,...,tm ) =exp E i∑ uk ξ(ω,tk ) Fϕ is conditional (with respect to F ) ϕ k=1 characteristic function ( Fϕ -characteristic function) of CLRP (1), which is expressed as follows: ∞ a ∈ , and if Eη(ω, τ) < ∞ then a = Eη(ω, τ) − ∫ xdG(x).
−∞
If random functions (fields) ϕ(ω, τ,t) and ϕ(ω, τ + s,t + s) are stochastically equivalent in the wide sense, that is, their finite-dimensional distributions are equal satisfying the following condition: n m n m P {ω : ϕ(ω, τi ,t j ) < xij } =P {ω : ϕ(ω, τi + s,t j + s) < xij }, xij ∈ ; (3) i=1 j=1 i=1 j=1 for any s ∈ , then conditional linear random process (1) is a strict sense stationary.
Using the above representation of m-dimensional characteristic function of conditional linear
random process the expressions for moment functions can be obtained which are important for
information signal processing (including mathematical expectation and covariance function), properties
of cyclostationarity [
In this section we define the general notion of continuous-time stationary ergodic random process.
We also analyze some cases which are important for the problems of information signal processing.
Finally, we define and prove the mixing property for conditional linear random process. Ergodicity is
the consequence of mixing property [
Let ξ(ω,t), t ∈ (−∞, ∞) is a continuous-time strictly stationary random process with the values in a measurable space {X ,B} and g(x1, x2 , ..., xm ) , m ≥ 1 is a Bm -measurable function satisfying the following condition:
Eg (ξ(ω,t1), ξ(ω,t2 ), ..., ξ(ω,tm )) < ∞, ∀t1,t2 , ...,tm ∈ . (4)
The continuous-time strictly stationary random process ξ(ω,t), t ∈ (−∞, ∞) is called ergodic if for any function g(x1, x2 , ..., xm ) satisfying the above conditions, the following holds with probability 1: 1 c lim ∫ g (ξ(ω,t1 + t), ξ(ω,t2 + t), ..., ξ(ω,tm + t)) dt = c→∞ c 0 (5) =Eg (ξ(ω,t1), ξ(ω,t2 ), ..., ξ(ω,tm )), ∀t1,t2 , ...,tm ∈ .
The above definition of ergodic random process is very general. But in the problems of information
signal analysis several cases are the most important. The mathematical expectation and covariance
function, one- and two-dimensional cumulative distribution functions and characteristic functions of
the investigated signal are usually estimated, analyzed, and used for informative features detection [
On the below expressions m is the dimension of the corresponding function g(x1, x2 , ..., xm ) ,
convergence is assumed with probability 1. Thus, we can define the following types of ergodicity of
continuous-time strictly stationary random process ξ(ω,t), t ∈ (−∞, ∞) [
Ergodicity with respect to the mathematical expectation Eξ(ω,t) =µ :
Fξ ( y) = P(ξ(ω,t) < y), y ∈ :
Ergodicity with respect to the covariance function R(τ) =E (ξ(ω,t) − µ)(ξ(ω,t + τ) − µ) , τ ∈ : =exp E i (u1ξ(ω,t) + u2ξ(ω,t + τ)) : 1 c m =1,g(x) =exp (iux) , t1 = 0 ⇒ lim ∫ exp(iuξ(ω,t))dt =fξ (u) .
c→∞ c 0 with respect to the two-dimensional characteristic function m = 2, g(x1, x2 ) =exp[i(u1x1 + u2 x2 )] , t1 = 0 , t2 = τ ⇒
1 c ⇒ lim ∫ exp i (u1ξ(ω,t) + u2ξ(ω,t + τ)) dt =fξ(u1,u2 ; τ) .
c→∞ c 0
It is easy to see that stationary random process can be ergodic with respect to mathematical expectation but not ergodic with respect to covariance function or characteristic function, etc.
The notion of ergodicity as it has been defined above represents the convergence of corresponding
sampling averages to probability characteristics of the random process. Mixing is another fundamental
property of random process expressed the fact that events (related to investigated process) separated by
long time intervals are approximately independent [
In terms of random process distributions, mixing property means that random vectors (ξ(ω,t1 + t), ξ(ω,t2 + t), ..., ξ(ω,tm + t)) and (ξ(ω, s1), ξ(ω, s2 ), ..., ξ(ω, sn )) (created by the samples of stationary random process) become approximately independent as t → ∞ .
A mixing property of the stationary random process represented by its characteristic functions is
defined as follows [
Ergodicity in the sense of (5) is a consequence of mixing. The hierarchy of mixing and ergodicity properties which are considered in the paper has been illustrated in Figure 1. Thus, to justify the ergodic properties of continuous-time strictly stationary conditional linear random process we need to prove the mixing property first. The idea is to use the characteristic function method using the representation (2) and relationship between conditional and unconditional characteristic functions. Obviously, the timedependent properties of the model expressed as stochastic integral driven by Levy process heavily depend on the corresponding properties of the random kernel.
Let ξ(ω,t), t ∈ (−∞, ∞) be a continuous-time strictly stationary conditional linear random process driven by Levy process, and with the kernel satisfying (3).
Let us denote
Law(ξ1(ω), ξ2 (ω), ..., ξm (ω)) =Law(η1(ω), η2 (ω), ..., ηm (ω)) if random vectors
(ξ1(ω), ξ2 (ω), ..., ξm (ω)) and (η1(ω), η2 (ω), ..., ηm (ω)) have the same m -dimensional cumulative distribution functions (distribution laws). ∀t1,t2 , ..., tm , s1, s2 , ..., sn ∈ . can be represented as:
Let random vectors
(ϕ(ω, τ,t1 + t), ϕ(ω, τ,t2 + t), ..., ϕ(ω, τ,tm + t)) and (ϕ(ω, τ, s1), ϕ(ω, τ, s2 ), ..., ϕ(ω, τ, sn )) are asymptotically independent as t → ∞ , ∀τ , t1,t2 , ..., tm , s1, s2 , ..., sn ∈ , that is, taking into account (3), the following condition holds: lim Law ( t →∞ ϕ(ω, τ + t,t1 + t), ϕ(ω, τ + t,t2 + t), ..., ϕ(ω, τ + t,tm + t), ϕ(ω, τ, s1), ϕ(ω, τ, s2 ), ..., ϕ(ω, τ, sn )) =
=Law (ϕ(ω, τ,t1), ϕ(ω, τ,t2 ), ..., ϕ(ω, τ,tm )) Law (ϕ(ω, τ, s1), ϕ(ω, τ, s2 ), ..., ϕ(ω, τ, sn )). ξ(ω,t) is the strictly stationary CLRP satisfying the mixing condition (6) (7) Indeed, (m + n) -dimensional Fϕ -characteristic function of the stationary CLRP with probability 1 fξFϕ (ω,u1,u2 , ..., um , v1, v2 , ..., vn ; t1 + t,t2 + t, ..., tm + t, s1, s2 , ..., sn ) =
m n =exp E i ∑k1 =ξ(ω,tk uk + t) + i ∑k1 =ξ(ω,sk νk ) Fϕ =
m ∞ n ∞ =ia exp ∑uk ∫ ϕ(ω, τ,tk + t)d τ + ∑ vk ∫ ϕ(ω, τ, sk )d τ +
k =−∞ 1 k =−∞ 1
∞ ∞ m n
+ ∫ ∫ exp ix ∑uk ϕ(ω, τ,tk + t) + ∑ vk ϕ(ω, τ, sk ) −1 −
−∞ −∞ k =1 k =1
ix ( ∑ =ϕ(ω,τ,tk uk + t) + ∑ =ϕ(ω,τ, vk sk ))
m n
k 1 k 1 ×
1 + x2
×
above function ψ(u) , and also properties of Fϕ -characteristic function [
t →∞ =Elti→m∞ fξFϕ (ω,u1,u2 , ..., um , v1, v2 , ..., vn ; t1 + t,t2 + t, ..., tm + t, s1, s2 , ..., sn ) = =(ω,u1,u2 E( fξFϕ , ..., um ; t1,t2 , ..., tm ) fξFϕ (ω, v1, v2 , ..., vn ; s1, s2 , ..., sn )) = =(ω,u1,u2 EfξFϕ , ..., um ; t1,t2 , ..., tm )EfξFϕ (ω, v1, v2 , ..., vn ; s1, s2 , ..., sn ) = = fξ (u1,u2 , ..., um ; t1,t2 , ..., tm ) fξ (v1, v2 , ..., vn ; s1, s2 , ..., sn ).
Thus, the condition (6) holds. That is, the investigated stationary conditional linear random process undergoes the mixing property. The ergodicity of the process in the sense of (5) is an immediate corollary of mixing property, including ergodicity with respect to moment functions, one- and multidimensional cumulative distribution functions and characteristic functions which are used for the feature detection and estimation in the problems of information signal processing.
The conditional linear random process driven by Levy process has been defined and described using conditional characteristic functions method, Levy-Khintchine form has been used to specify the Poisson jump spectrum of Levy process. The CLRP belongs to the class of infinitely divisible mixtures. The condition of CLRP to be strict sense stationary has been represented.
The hierarchy of mixing and ergodicity properties has been analyzed, the importance of the corresponding concepts for the problems of applied information signal modelling and processing has been represented. Thus, the effective mathematical model should undergo ergodicity and mixing properties.
The mixing property of the continuous-time strictly stationary conditional linear random process has been proven using the characteristic functions method. The ergodicity property is a consequence of mixing. That is why, if the information signal is modelled as stationary ergodic CLRP then it is priori justification of performing the statistical analysis using time averaging of its single realization.
The prospective research is related to the study of mixing and ergodicity of discrete-time conditional linear random process.