### 1 Multiple Description Coding for Stationary Gaussian Sources

2 for stationary Gaussian sources is still unknown. The main contribution of this work is an exact characterization of the rate region for the 2-description case. The remainder of

### in a stationary gaussian process - American Mathematical ...

From the discussion at the end of §4 of [6], it is clear that the strong mixing condition is sufficient for the result of Theorem 2.1. 3. The almost sure behaviour.

### LONG RANGE DEPENDENCE 1. Introduction Long range ...

sufficient conditions for strong mixing of a stationary Gaussian process were later established in Helson and Sarason (1967). Explicit necessary and suffi-.

### Simulating a class of stationary Gaussian processes using ...

Jean-Francois Coeurjolly, Emilio Porcu, Fast and Exact Simulation of Complex-Valued Stationary Gaussian Processes Through Embedding Circulant Matrix, Journal of Computational and Graphical Statistics, 10.1080/10618600.2017.1385468, 27, 2, (278-290), (2017).

### On Modeling a Class of Weakly Stationary Processes - Frontiers

2020年1月15日 — Limit theorems for strong mixing processes are studied (e.g., [3–5]). However, specific mixing conditions are often more than difficult to verify. ... (Xt)t∈ℕ is a d-variate stationary Gaussian process and f is a given function.

### Lecture 13 Time Series: Stationarity, AR(p) & MA(q)

A: We need to impose conditions on ρk. Conditions weaker than "they are all zero;" but, strong enough to exclude the sequence of identical copies. Time Series – Ergodicity of the Mean • Definition: A covariance-stationary process is ergodic for the mean if plimz E(Zt) Ergodicity Theorem: Then, a sufficient condition for ergodicity for

### THE SPECTRAL DENSITY OF A STRONGLY MIXING STATIONARY GAUSSIAN ...

Gaussian processes which satisfy a certain strong mixing condition. Helson and Sarason studied the analogous class of weights on the unit circle, which correspond to discrete-time processes.

### Asymptotics of Maxima of Strongly Dependent Gaussian …

Key Words: Stationary Gaussian process; strong dependence; Berman condition; limit theorems; Pickands constant. AMS Classi cation: primary 60G15; secondary 60G70 1 Introduction Let fX(t);t2[0;1)gbe a standard (mean zero and unit variance) stationary Gaussian process with continuous sample paths, and let fr(t);t 0gdenote its correlation function.

### Some mixing conditions for stationary symmetric stable ...

We derive some necessary and sufficient conditions for mixing of non-Gaussian stationary symmetric stable processes in terms of the spectral representation, and derive additional conditions for the special case where the spectral representation itself is stationary.

### Lecture 1: Stationary Time Series

Because if a process is Gaussian, uncorrelation implies independence. Therefore, a Gaussian white noise is just i.i.d.N(0,σ2). Stationary and nonstationary processes are very diﬀerent in their properties, and they require diﬀerent inference procedures. We will discuss this in much details through this course. At this point, note that a ...

### Asymptotic theory for time-series with cyclic and trend ...

ones by relaxing the strong mixing conditions of the noise process and putting some mild ... non-stationary processes and can be considered as a bridge between the stationary and ... is obtained using invariance principle and extreme value theory for Gaussian processes. 3.1. …

### Basic Properties of Strong Mixing Conditions. A Survey and ...

by the author on basic properties of strong mixing conditions. AMS 2000 subject classiﬁcations: Primary 60G10. Keywords and phrases: strong mixing conditions, stationary sequences. Received April 2005. This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions.

### Maxima and High Level Excursions of Stationary Gaussian Processes

conditions (0.6) and (0.7), assumed that the process satisfies the "strong mixing condition." Their proof is based on the fact that the "horizontal-window" con-ditional limiting distribution of the excursion above a high level is T2, the Rayleigh distribution; and that the upcrossings tend to a limiting Poisson process.

### Nonlinear ICA of Temporally Dependent Stationary Sources

Aug 18, 2020 · We prove that the method estimates the sources for general smooth mixing nonlinearities, assuming the sources have sufficiently strong temporal dependencies, and these dependencies are in a certain way different from dependencies found in Gaussian processes. For Gaussian (and similar) sources, the method estimates the nonlinear part of the mixing.

### On Modeling a Class of Weakly Stationary Processes - Aalto ...

2020年1月1日 — conditions. Limit theorems for strong mixing processes are studied (e.g., [3–5]). However, specific mixing conditions are often more than difficult to verify. ... Gaussian subordinated processes in modeling weakly stationary.

### Asymptotics of Maxima of Strongly Dependent Gaussian Processes

In this section, we extend Theorem A to a sequence of strongly dependent stationary Gaussian processes. A sequence of standard stationary Gaussian process fX n(t);t2[0;1)g;n2N is called strongly dependent if the correlation function r n(t) satis es one of the following assumptions: (B1). r n(t)logt!r2(0;1) as t!1, uniformly in n; (B2). r

### Asymptotic Distribution Of Sum And Maximum For Strongly ...

The main result in Hsing (1995) is that for strong mixing sequences, such that Sn satisfies the central limit theorem, asymptotic independence of (Sn, Mn) ensues. Gaussian sequences have long been studied with regard to the asymptotic properties of extreme values. It is well known that for stationary Gaussian sequences 6 Xn > with E Xn = 0 and E Xn

### Basic Properties of Strong Mixing Conditions. A Survey ... - arXiv

2005年11月3日 — Keywords and phrases: strong mixing conditions, stationary sequences. Received ... For mixing properties of linear processes, see [71] and [143]. ... For stationary Gaussian random fields, a version of (2) had been proved.

### On Strong Mixing Conditions for Stationary Gaussian Processes

This paper considers conditions, which guarantee strong mixing of stationary random Gaussian process $\xi (t)$. It is proved, for example, that if the spectral ...

### (PDF) Strong mixing coefficients for non-commutative ...

Strong mixing coefficients for non-commutative Gaussian processes Article (PDF Available) in Proceedings of the American Mathematical Society 132(2) · July 2003 with 16 Reads How we measure 'reads'

### On Strong Mixing Conditions for Stationary Gaussian Processes ...

Jul 28, 2006 · (1965) On The Spectrum Of Stationary Gaussian Sequences Satisfying the Strong Mixing Condition I. Necessary Conditions. Theory of Probability & Its Applications 10 :1, 85-106. Citation | PDF (1412 KB)

### A strong mixing condition for second-order stationary ... - EUDML

A strong mixing condition for second-order stationary random fields ... The spectral density of a strongly mixing stationary Gaussian process, Pacific J. Math.

### On Modeling a Class of Weakly Stationary Processes

consists of stationary processes (zt)t∈N of the form zt = f(Xt), where (Xt)t∈N is a d-variate stationary Gaussian process and f is a given function. It is usually assumed that f(X0) ∈ L2. Central limit theorems for such time series date back to Breuer and Major [12] and the topic has been studied extensively. Indeed, for Gaussian ...

### functionals of stationary Gaussian processes arXiv:math ...

2. Crossings of Gaussian processes Studies on level-crossings by stationary Gaussian processes began about sixty years ago. Diﬀerent approaches have been proposed. Here is a survey of the literature on the number of crossings of a given level or of a diﬀerentiable curve in a ﬁxed time interval by a continuous spectrum Gaussian process.

### Asymptotic Theory for Estimators of High-order Statistics ...

of stationary processes. Gaussian processes are completely characterized by their rst and second-order moments. However many processes observed in practice are non-Gaussian. To study the non-Gaussianaity of a process, one should go beyond second-order statistics and consider high-order moments or cumulants. High-order cumulants can contain the

### 1. STATIONARY GAUSSIAN PROCESSES - ERNET

The process X is called stationary (or translation invariant) if Xτ =d X for all τ∈T. Let X be a Gaussian process on T with mean M: T → R and covariance K: T ×T → R. It is an easy exercise to see that X is stationary if and only if M is a constant and K(t,s) depends only ont−s. In this case we usually write the covariance as K(t−s ...

### What is dependence? TWO papers provide background: 1 ...

gree overcome some drawbacks of strong mixing conditions. In many cases it is not ... asserts that any weakly stationary process can be decom- posed into a ...

### Stationary Sequences and Random Fields ...

A derivation of the asymptotic distribution for spectral (second order) estimates is given under an assumption of strong mixing in Chapter V. A ... The results will be obtained for complex-valued weakly stationary processes. ... Quadratic Forms, Limit Theorems and Mixing Conditions ... Non-Gaussian Linear Processes.