A maximal law of the iterated logarithm for operator-normalized stochastically compact partial sums of i.i.d. random vectors

1985 ◽  
pp. 426-439 ◽  
Author(s):  
Daniel Charles Weiner
Keyword(s):  
Partial Sums ◽  
Random Vectors ◽  
Iterated Logarithm

Law of the iterated logarithm for self-normalized sums and their increments

2006 ◽  
Vol 43 (1) ◽  
pp. 79-114
Author(s):  
Han-Ying Liang ◽  
Jong-Il Baek ◽  
Josef Steinebach
Keyword(s):  
Random Variables ◽  
Domain Of Attraction ◽  
Weighted Sums ◽  
Partial Sums ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Independent And Identically Distributed

Let X1, X2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses Mn=max 1?i?n|Xi| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

scholarly journals The law of the iterated logarithm on subsequences-characterizations

1990 ◽  
Vol 118 ◽  
pp. 65-97 ◽  
Author(s):  
Michel Weber
Keyword(s):  
Unit Ball ◽  
Covariance Matrix ◽  
Identity Matrix ◽  
Random Vectors ◽  
Unbounded Function ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Euclidian Space ◽  
Zero Vector ◽  
Increasing Sequences

Let be any increasing sequence of integers and M> 1; we connect to them in a very simply way, an increasing unbounded function φ: → R+. Let also X1, X2, · · · be a sequence of i.i.d. random vectors with value in euclidian space Rm. We prove that the cluster set of the sequence almost surely coincides with the unit ball of Rm, if, and only if, the covariance matrix of X1 is the identity matrix of Rm and EX1 is the zero vector of Rm. We define a functional A on the set of increasing sequences of integers as follows:.

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

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
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)

On Bernstein Type and Maximal Inequalities for Dependent Banach-Valued Random Vectors and Applications

2018 ◽  
Author(s):  
Noureddine Rhomari
Keyword(s):  
Recursive Estimation ◽  
Covariance Operator ◽  
Partial Sums ◽  
Strong Mixing ◽  
Random Vectors ◽  
Bernstein Type ◽  
Strong Law ◽  
Maximal Inequalities ◽  
Mixing Coefficients ◽  
Measure Of Dependence

This article discusses some results on Bernstein type and maximal inequalities for partial sums of dependent random vectors taking their values in separable Hilbert or Banach spaces of finite or infinite dimension. Two types of measure of dependence are considered: strong mixing coefficients (α-mixing) and absolutely regular mixing coefficients (β-mixing). These inequalities, which are similar to those in the dependent real case, are used to derive the strong law of large numbers (SLLN) and the bounded law of the iterated logarithm (LIL) for absolutely regular Hilbert- or Banach-valued processes under minimal mixing conditions. The article first introduces the relevant notation and definitions before presenting the maximal inequalities in the strong mixing case, followed by the absolutely regular mixing case. It concludes with some applications to the SLLN, the bounded LIL for Hilbertian or Banachian absolutely regular processes, the recursive estimation of probability density, and the covariance operator estimations.

Ordering Functions of Random Vectors, with Application to Partial Sums

2012 ◽  
Vol 26 (2) ◽  
pp. 474-479 ◽  
Author(s):  
Michel M. Denuit ◽  
Mhamed Mesfioui
Keyword(s):  
Partial Sums ◽  
Random Vectors

Limsup theorems and a uniform LIL for asymptotically negatively associated random sequences

2012 ◽  
Vol 49 (2) ◽  
pp. 223-235
Author(s):  
Yong-Kab Choi ◽  
Kyo-Shin Hwang
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Random Sequences ◽  
Negatively Associated ◽  
Iterated Logarithm ◽  
Uniform Law

In this paper we establish some limsup theorems and a generalized uniform law of the iterated logarithm (LIL) for the increments of the partial sums of a strictly stationary and asymptotically negatively associated (ANA) sequence of random variables.

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