Extensions and Complements

This chapter contains treatments of a range of topics associated with the central limit theorem. These include estimated normalization using methods of heteroscedasticity and autocorrelation consistent variance estimation, the CLT in linear prrocesses, random norming giving rise to a mixed Gaussian limiting distribution, and the Cramér–Wold device and multivariate CLT. The delta method to derive the limit distributions of differentiable functions is described. The law of the iterated logarithm is proved for Gaussian processes.

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 ◽

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.

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 ◽

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 ◽

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)

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 ◽

Variance estimation in the central limit theorem for Markov chains

Journal of Statistical Planning and Inference ◽

10.1016/j.jspi.2008.10.020 ◽

2009 ◽

Vol 139 (7) ◽

pp. 2242-2253 ◽

Cited By ~ 5

Author(s):

Samis Trevezas ◽

Nikolaos Limnios

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Markov Chains ◽

Central Limit ◽

Variance Estimation ◽

The Central Limit Theorem

On the central limit theorem and the law of the iterated logarithm

Statistics & Probability Letters ◽

10.1016/j.spl.2007.12.014 ◽

2008 ◽

Vol 78 (12) ◽

pp. 1384-1387 ◽

Cited By ~ 1

Author(s):

Katusi Fukuyama ◽

Yôhei Ueno

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Iterated Logarithm ◽

The Law

The central limit theorem and the law of iterated logarithm for empirical processes under local conditions

Probability Theory and Related Fields ◽

10.1007/bf00334041 ◽

1988 ◽

Vol 77 (2) ◽

pp. 271-305 ◽

Cited By ~ 22

Author(s):

Niels T. Andersen ◽

Evarist Giné ◽

Mina Ossiander ◽

Joel Zinn

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Empirical Processes ◽

Law Of Iterated Logarithm ◽

Local Conditions ◽

Iterated Logarithm ◽

The Law

Some Iterated Logarithm Results Related to the Central Limit Theorem

Transactions of the American Mathematical Society ◽

10.2307/1995607 ◽

1971 ◽

Vol 156 ◽

pp. 185

Author(s):

R. J. Tomkins

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

On a Relation Between an Estimate of the Remainder in the Central Limit Theorem and the Law of the Iterated Logarithm

Theory of Probability and Its Applications ◽

10.1137/1111046 ◽

1966 ◽

Vol 11 (3) ◽

pp. 454-458 ◽

Cited By ~ 8

Author(s):

V. V. Petrov

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Iterated Logarithm ◽

The Law

STATISTICAL LIMIT LAWS FOR EQUIVARIANT OBSERVATIONS

Stochastics and Dynamics ◽

10.1142/s0219493704000882 ◽

2004 ◽

Vol 04 (01) ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

IAN MELBOURNE ◽

MATTHEW NICOL

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Probability Space ◽

Limit Laws ◽

Weak Invariance Principle ◽

Weak Invariance ◽

Iterated Logarithm ◽

Statistical Limit

We show that statistical limit laws for ergodic stationary sequences of G-equivariant observations ϕ on a probability space Ω×G are inherited by sequences of observations ϕ(·,g0) on the probability space Ω for each fixed g0. The statistical limit laws we consider are the central limit theorem, weak invariance principle and the law of the iterated logarithm.