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

2002 ◽  
Vol 39 (3-4) ◽  
pp. 333-359 ◽  
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).

COMPLETE CONVERGENCE FOR ARRAYS AND THE LAW OF THE SINGLE LOGARITHM

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.

Limit theorems for weakly exchangeable arrays

1978 ◽  
Vol 84 (1) ◽  
pp. 123-130 ◽  
Author(s):  
G. K. Eagleson ◽  
N. C. Weber
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Moment Conditions ◽  
Large Numbers ◽  
Independence Structure ◽  
Symmetry Relations

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

Some limit theorems for a subcritical branching process by immigration

1984 ◽  
Vol 21 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Y. S. Chow ◽  
K. F. Yu
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Total Population ◽  
Strong Law ◽  
Moment Conditions ◽  
Moment Convergence ◽  
Large Numbers ◽  
The Moment ◽  
Branching Process With Immigration

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

scholarly journals Strong Limit Theorems for Dependent Random Variables

2021 ◽  
Vol 15 ◽  
pp. 15-17
Author(s):  
Libin Wu ◽  
Bainian Li
Keyword(s):  
Limit Theorems ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Moment Inequality ◽  
Strong Limit ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Independent And Identically Distributed

In this article We establish moment inequality of dependent random variables, furthermore some theorems of strong law of large numbers and complete convergence for sequences of dependent random variables. In particular, independent and identically distributed Marcinkiewicz Law of large numbers are generalized to the case of m₀ -dependent sequences.

scholarly journals Limit theorems for fuzzy random variables

1986 ◽  
Vol 407 (1832) ◽  
pp. 171-182 ◽  
Keyword(s):  
Banach Space ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Embedding Theorems ◽  
Fuzzy Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Fuzzy Random

A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.

Some limit theorems for a subcritical branching process by immigration

1984 ◽  
Vol 21 (01) ◽  
pp. 50-57
Author(s):  
Y. S. Chow ◽  
K. F. Yu
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Total Population ◽  
Strong Law ◽  
Moment Conditions ◽  
Moment Convergence ◽  
Large Numbers ◽  
The Moment ◽  
Branching Process With Immigration

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

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