Limit theorems for locally stationary processes

AbstractThe purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

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COMPLETE CONVERGENCE FOR ARRAYS AND THE LAW OF THE SINGLE LOGARITHM

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717000582 ◽

2017 ◽

Vol 96 (2) ◽

pp. 333-344

Author(s):

ALLAN GUT ◽

ULRICH STADTMÜLLER

Keyword(s):

Limit Theorems ◽

Complete Convergence ◽

Law Of Large Numbers ◽

Almost Sure Convergence ◽

Strong Law ◽

Moment Conditions ◽

Large Numbers ◽

Distributed Arrays ◽

The Law ◽

Independent And Identically Distributed

The present paper is devoted to complete convergence and the strong law of large numbers under moment conditions near those of the law of the single logarithm (LSL) for independent and identically distributed arrays. More precisely, we investigate limit theorems under moment conditions which are stronger than $2p$ for any $p<2$, in which case we know that there is almost sure convergence to 0, and weaker than $E\,X^{4}/(\log ^{+}|X|)^{2}<\infty$, in which case the LSL holds.

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Limit theorems for sums of chain-dependent processes

Journal of Applied Probability ◽

10.2307/3212704 ◽

1974 ◽

Vol 11 (3) ◽

pp. 582-587 ◽

Cited By ~ 19

Author(s):

G. L. O'Brien

Keyword(s):

Markov Chain ◽

Limit Theorem ◽

Limit Theorems ◽

Law Of Large Numbers ◽

Initial Distribution ◽

Stationary Case ◽

Strong Law ◽

Iterated Logarithm ◽

Large Numbers ◽

Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

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On the Dependence of the Convergence Rate in the Strong Law of Large Numbers for Stationary Processes on the Rate of Decay of the Correlation Function

Theory of Probability and Its Applications ◽

10.1137/1126078 ◽

1982 ◽

Vol 26 (4) ◽

pp. 706-720 ◽

Cited By ~ 6

Author(s):

V. F. Gaposhkin

Keyword(s):

Correlation Function ◽

Convergence Rate ◽

Law Of Large Numbers ◽

Stationary Processes ◽

Strong Law ◽

Large Numbers ◽

Rate Of Decay

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A REMARK ON THE STRONG LAW FOR B-VALUED ARRAYS OF RANDOM ELEMENTS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972710000158 ◽

2010 ◽

Vol 82 (1) ◽

pp. 31-43 ◽

Cited By ~ 1

Author(s):

TIEN-CHUNG HU ◽

PING YAN CHEN ◽

N. C. WEBER

Keyword(s):

Law Of Large Numbers ◽

Moving Average ◽

Average Process ◽

Moving Average Process ◽

Strong Law ◽

Rate Law ◽

Large Numbers ◽

Random Elements

AbstractThe conditions in the strong law of large numbers given by Li et al. [‘A strong law for B-valued arrays’, Proc. Amer. Math. Soc.123 (1995), 3205–3212] for B-valued arrays are relaxed. Further, the compact logarithm rate law and the bounded logarithm rate law are discussed for the moving average process based on an array of random elements.

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One-sided strong laws forincrements of sumsof i.i.d. random variables

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.39.2002.3-4.6 ◽

2002 ◽

Vol 39 (3-4) ◽

pp. 333-359 ◽

Cited By ~ 2

Author(s):

A. N. Frolov

Keyword(s):

Limit Theorems ◽

Strong Approximation ◽

Law Of Large Numbers ◽

Random Variables ◽

Strong Law ◽

Moment Conditions ◽

Second Moments ◽

Large Numbers ◽

Norming Sequence ◽

Normal Law

We find a universal norming sequence in strong limit theorems for increments of sums of i.i.d. random variables with finite first moments and finite second moments of positive parts. Under various one-sided moment conditions our universal theorems imply the following results for sums and their increments: the strong law of large numbers, the law of the iterated logarithm, the Erdős-Rényi law of large numbers, the Shepp law, one-sided versions of the Csörgő-Révész strong approximation laws. We derive new results for random variables from domains of attraction of a normal law and asymmetric stable laws with index αЄ(1,2).

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Limit theorems for sums of chain-dependent processes

Journal of Applied Probability ◽

10.1017/s0021900200096388 ◽

1974 ◽

Vol 11 (03) ◽

pp. 582-587 ◽

Cited By ~ 5

Author(s):

G. L. O'Brien

Keyword(s):

Markov Chain ◽

Limit Theorem ◽

Limit Theorems ◽

Law Of Large Numbers ◽

Initial Distribution ◽

Stationary Case ◽

Strong Law ◽

Iterated Logarithm ◽

Large Numbers ◽

Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

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