scholarly journals Asymptotic normality of recursive estimators under strong mixing conditions

2013 ◽  
Vol 16 (2) ◽  
pp. 81-96 ◽  
Author(s):  
Aboubacar Amiri
Keyword(s):  
Asymptotic Normality ◽  
Strong Mixing ◽  
Mixing Conditions ◽  
Recursive Estimators

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 Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property

2021 ◽  
Vol 5 (1) ◽  
pp. 19
Author(s):  
Alexander Kushnir ◽  
Alexander Varypaev
Keyword(s):  
Time Series ◽  
Asymptotic Normality ◽  
Multivariate Time Series ◽  
Asymptotic Properties ◽  
Asymptotic Distributions ◽  
Stationary Time Series ◽  
Strong Mixing ◽  
Statistical Estimates ◽  
Distribution Parameters ◽  
M Estimates

The publication is devoted to studying asymptotic properties of statistical estimates of the distribution parameters u∈Rq of a multidimensional random stationary time series zt∈Rm, t∈ℤ satisfying the strong mixing conditions. We consider estimates u^nδ(z¯n), z¯n=(z1T,…,znT)T∈Rmn that provide in asymptotic n→∞ the maximum values for some objective functions Qn(z¯n;u), which have properties similar to the well-known property of local asymptotic normality. These estimates are constructed by solving the equations δn(z¯n;u)=0, where δn(z¯n;u) are arbitrary functions for which δn(z¯n;u)−gradhQn(z¯n;u+n−1/2h)→0(n→∞) in Pn,u(z¯n)-probability uniformly on u∈U, were U is compact in Rq. In many cases, the estimates u^nδ(z¯n) have the same asymptotic properties as well-known M-estimates defined by equations u^nQ(z¯n)=arg maxu∈UQn(z¯n;u) but often can be much simpler computationally. We consider an algorithmic method for constructing estimates u^nδ(z¯n), which is similar to the accumulation method first proposed by R. Fischer and rigorously developed by L. Le Cam. The main theoretical result of the article is the proof of the theorem, in which conditions of the asymptotic normality of estimates u^nδ(z¯n) are formulated, and the expression is proposed for their matrix of asymptotic mean-square deviations limn→∞nEn,u{(u^δ(z¯n)−u)(u^δ(z¯n)−u)T}.

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.

scholarly journals Conditional risk estimate for functional data under strong mixing conditions

2015 ◽  
Vol 14 (3) ◽  
pp. 301
Author(s):  
Abbes Rabhi ◽  
Sara Soltani ◽  
Aboubacar Traore
Keyword(s):  
Functional Data ◽  
Risk Estimate ◽  
Strong Mixing ◽  
Mixing Conditions ◽  
Conditional Risk
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