Some remarks on strong mixing conditions

1985 ◽  
pp. 65-72
Keyword(s):  
Strong Mixing ◽  
Mixing Conditions

ON THE SPECTRAL MEASURES OF SOME WEAKLY STATIONARY SEQUENCES INVOLVING RANDOMLY SPACED OBSERVATIONS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150004
Author(s):  
RICHARD C. BRADLEY
Keyword(s):  
Spectral Density ◽  
Density Function ◽  
Spectral Density Function ◽  
Strong Mixing ◽  
Spectral Measures ◽  
Mixing Conditions ◽  
Stationary Sequences ◽  
Second Moments ◽  
The Central Limit Theorem ◽  
Complex Valued

In an earlier paper by the author, as part of a construction of a counterexample to the central limit theorem under certain strong mixing conditions, a formula is given that shows, for strictly stationary sequences with mean zero and finite second moments and a continuous spectral density function, how that spectral density function changes if the observations in that strictly stationary sequence are "randomly spread out" in a particular way, with independent "nonnegative geometric" numbers of zeros inserted in between. In this paper, that formula will be generalized to the class of weakly stationary, mean zero, complex-valued random sequences, with arbitrary spectral measure.

scholarly journals Bernstein-type inequality for a class of dependent random matrices

2016 ◽  
Vol 05 (02) ◽  
pp. 1650006 ◽  
Author(s):  
Marwa Banna ◽  
Florence Merlevède ◽  
Pierre Youssef
Keyword(s):  
Random Matrices ◽  
Type Inequality ◽  
Bernstein Inequality ◽  
Mathematical Statistics ◽  
Strong Mixing ◽  
Mixing Conditions ◽  
Bernstein Type ◽  
Bernstein Type Inequality ◽  
The Matrix ◽  
The Laplace Transform

In this paper, we obtain a Bernstein-type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. [Bernstein inequality and moderate deviations under strong mixing conditions, in High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics Collection, Vol. 5 (Institute of Mathematical Statistics, Beachwood, OH, 2009), pp. 273–292.] in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.

Non-strong mixing autoregressive processes

1984 ◽  
Vol 21 (4) ◽  
pp. 930-934 ◽  
Author(s):  
Donald W. K. Andrews
Keyword(s):  
Direct Proof ◽  
Strong Mixing ◽  
Autoregressive Processes ◽  
Mixing Conditions ◽  
First Order ◽  
Insight Into

Certain first-order autoregressive processes are shown not to be strong mixing. A direct proof is given. This proof gives considerably more insight into the nature of the result than do proofs by contradiction. The result and proof help to clarify the relation between the autoregressive and strong mixing conditions.

Nonparametric Methods for α-Mixing Functional Random Variables

2018 ◽  
Author(s):  
Laurent Delsol
Keyword(s):  
Kernel Smoothing ◽  
Asymptotic Properties ◽  
Strong Mixing ◽  
Kernel Estimators ◽  
Mixing Conditions ◽  
Exponential Inequalities ◽  
Smoothing Methods ◽  
Mixing Coefficients ◽  
Mixing Sequences ◽  
Prediction Problems

This article considers how functional kernel methods can be used to study α-mixing datasets. It first provides an overview of how prediction problems involving dependent functional datasets may arise from the study of time series, focusing on the standard discretized model and modelization that takes into account the functional nature of the evolution of the quantity to be studied over time. It then considers strong mixing conditions, with emphasis on the notion of α-mixing coefficients and α-mixing variables introduced by Rosenblatt (1956). It also describes some conditions for a Markov chain to be α-mixing; some useful tools that provide covariance inequalities, exponential inequalities, and Central Limit Theorem (CLT) for α-mixing sequences; the asymptotic properties of functional kernel estimators; the use of kernel smoothing methods with α-mixing datasets; and various functional kernel estimators corresponding to different prediction methods. Finally, the article highlights some interesting prospects for further research.

Non-strong mixing autoregressive processes

1984 ◽  
Vol 21 (04) ◽  
pp. 930-934 ◽  
Author(s):  
Donald W. K. Andrews
Keyword(s):  
Direct Proof ◽  
Strong Mixing ◽  
Autoregressive Processes ◽  
Mixing Conditions ◽  
First Order ◽  
Insight Into

Certain first-order autoregressive processes are shown not to be strong mixing. A direct proof is given. This proof gives considerably more insight into the nature of the result than do proofs by contradiction. The result and proof help to clarify the relation between the autoregressive and strong mixing conditions.

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