On the second Borel-Cantelli lemma for strongly mixing sequences of events

1997 ◽  
Vol 34 (02) ◽  
pp. 381-394 ◽  
Author(s):  
Dirk Tasche
Keyword(s):  
Law Of Large Numbers ◽  
Exponential Rate ◽  
Cantelli Lemma ◽  
Mixing Rate ◽  
Strong Law ◽  
Moment Conditions ◽  
Sequence Of Events ◽  
Large Numbers ◽  
Strongly Mixing ◽  
Mixing Sequences

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.

On the second Borel-Cantelli lemma for strongly mixing sequences of events

1997 ◽  
Vol 34 (2) ◽  
pp. 381-394 ◽  
Author(s):  
Dirk Tasche
Keyword(s):  
Law Of Large Numbers ◽  
Exponential Rate ◽  
Cantelli Lemma ◽  
Mixing Rate ◽  
Strong Law ◽  
Moment Conditions ◽  
Sequence Of Events ◽  
Large Numbers ◽  
Strongly Mixing ◽  
Mixing Sequences

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.

scholarly journals A NONCONVENTIONAL STRONG LAW OF LARGE NUMBERS AND FRACTAL DIMENSIONS OF SOME MULTIPLE RECURRENCE SETS

2012 ◽  
Vol 12 (03) ◽  
pp. 1150023 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Law Of Large Numbers ◽  
Fractal Dimensions ◽  
Growth Conditions ◽  
Multiple Recurrence ◽  
Strong Law ◽  
Moment Conditions ◽  
Vector Process ◽  
Regularity Properties ◽  
Large Numbers ◽  
Positive Functions

We provide conditions which yield a strong law of large numbers for expressions of the form [Formula: see text] where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and qi, i > m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q1(n), …, qℓ(n).

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.

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).

scholarly journals Strong law of large numbers forρ*-mixing sequences with different distributions

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Guang-Hui Cai
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Mixing Sequences

Strong law of large numbers and complete convergence forρ*-mixing sequences with different distributions are investigated. The results obtained improve the relevant results by Utev and Peligrad (2003).

Strong Laws for Dependent Heterogeneous Processes

1991 ◽  
Vol 7 (2) ◽  
pp. 213-221 ◽  
Author(s):  
Bruce E. Hansen
Keyword(s):  
Law Of Large Numbers ◽  
Strong Mixing ◽  
Strong Law ◽  
Strong Laws ◽  
Maximal Inequalities ◽  
Large Numbers ◽  
Heterogeneous Processes ◽  
Mixing Sequences ◽  
Strong Mixing Sequences ◽  
Weakly Dependent

This paper presents maximal inequalities and strong law of large numbers for weakly dependent heterogeneous random variables. Specifically considered are Lr mixingales for r > 1, strong mixing sequences, and near epoch dependent (NED) sequences. We provide the first strong law for Lr-bounded Lr mixingales and NED sequences for 1 > r > 2. The strong laws presented for α-mixing sequences are less restrictive than the laws of McLeish [8].

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.

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