On the central limit theorem and iterated logarithm law for stationary processes

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.

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Improved classical limit analogues for Galton-Watson processes with or without immigration

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700047018 ◽

1971 ◽

Vol 5 (2) ◽

pp. 145-155 ◽

Cited By ~ 18

Author(s):

C.C. Heyde ◽

J.R. Leslie

Keyword(s):

Random Walk ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Classical Limit ◽

Moment Conditions ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

Logarithm Law ◽

Iterated Logarithm Law

It has recently emerged that the central limit theorem and iterated logarithm law for random walk processes have natural counterparts for Galton-Watson processes with or without immigration. Much of the work on these counterparts has previously involved the imposition of supplementary moment conditions. In this paper we show how to dispense with these supplementary conditions and in so doing make the analogy with the random walk results complete.

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Some limit theorems for Markov branching processes

Journal of Applied Probability ◽

10.1017/s0021900200095309 ◽

1973 ◽

Vol 10 (02) ◽

pp. 299-306 ◽

Cited By ~ 1

Author(s):

J. R. Leslie

Keyword(s):

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Original Process ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

Logarithm Law ◽

Increasing Process ◽

Watson Process ◽

Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

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On the central limit theorem and law of the iterated logarithm for stationary processes with applications to linear processes

Stochastic Processes and their Applications ◽

10.1016/0304-4149(95)00038-9 ◽

1995 ◽

Vol 59 (2) ◽

pp. 343-351 ◽

Cited By ~ 5

Author(s):

Ryozo Yokoyama

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Stationary Processes ◽

Linear Processes ◽

The Central Limit Theorem ◽

Iterated Logarithm

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Some limit theorems for Markov branching processes

Journal of Applied Probability ◽

10.2307/3212347 ◽

1973 ◽

Vol 10 (2) ◽

pp. 299-306 ◽

Cited By ~ 3

Author(s):

J. R. Leslie

Keyword(s):

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Original Process ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

Logarithm Law ◽

Increasing Process ◽

Watson Process ◽

Iterated Logarithm Law

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ON THE RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR TWO-PARAMETER MARTINGALE DIFFERENCES

Acta Mathematica Scientia ◽

10.1016/s0252-9602(17)30804-4 ◽

1996 ◽

Vol 16 (3) ◽

pp. 287-295

Author(s):

Hongwei Long

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Rates Of Convergence ◽

Martingale Differences ◽

The Central Limit Theorem ◽

Two Parameter

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Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

Mathematical Problems in Engineering ◽

10.1155/2021/6691857 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Mingzhou Xu ◽

Kun Cheng

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Uniform Convergence ◽

Central Limit ◽

Random Variables ◽

Precise Asymptotics ◽

Partial Sum ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

The Law

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.

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Invariance Principles in the Central Limit Theorem and Law of the Iterated Logarithm

Martingale Limit Theory and its Application ◽

10.1016/b978-0-12-319350-6.50010-4 ◽

1980 ◽

pp. 97-125

Author(s):

P. Hall ◽

C.C. Heyde

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Invariance Principles ◽

The Central Limit Theorem ◽

Iterated Logarithm

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On Feller's criterion for the law of the iterated logarithm

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171294000463 ◽

1994 ◽

Vol 17 (2) ◽

pp. 323-340 ◽

Cited By ~ 1

Author(s):

Deli Li ◽

M. Bhaskara Rao ◽

Xiangchen Wang

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Uniform Estimate ◽

Random Variables ◽

Independent Random Variables ◽

Partial Sums ◽

The Central Limit Theorem ◽

Iterated Logarithm ◽

The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

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On Gordin's central limit theorem for stationary processes

Journal of Applied Probability ◽

10.2307/3212425 ◽

1975 ◽

Vol 12 (1) ◽

pp. 176-179 ◽

Cited By ~ 7

Author(s):

G. K. Eagleson

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Stationary Processes ◽

The Central Limit Theorem ◽

Mixture Of Normals

The central limit theorem for ergodic stationary processes obtained by Gordin is shown to hold for general stationary processes. In this case, the limit law is a mixture of normals.

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