scholarly journals Limit theorems for pure death processes coming down from infinity

2017 ◽  
Vol 54 (3) ◽  
pp. 720-731 ◽  
Author(s):  
Serik Sagitov ◽  
Thibaut France
Keyword(s):  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Natural Generalization ◽  
Death Process ◽  
Strong Law ◽  
Large Numbers ◽  
Large Deviation Theorem

Abstract In this paper we treat a pure death process coming down from infinity as a natural generalization of the death process associated with the Kingman coalescent. We establish a number of limit theorems including a strong law of large numbers and a large deviation theorem.

SOME LIMIT THEOREMS OF DELAYED AVERAGES FOR COUNTABLE NONHOMOGENEOUS MARKOV CHAINS

2019 ◽  
pp. 1-19
Author(s):  
Pingping Zhong ◽  
Weiguo Yang ◽  
Zhiyan Shi ◽  
Yan Zhang
Keyword(s):  
Information Theory ◽  
Markov Chains ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Strong Ergodicity ◽  
Large Numbers ◽  
Nonhomogeneous Markov Chains ◽  
Definition Of ◽  
Bivariate Functions

AbstractThe purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

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 sums of chain-dependent processes

1974 ◽  
Vol 11 (3) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

scholarly journals The Strong Law of Large Numbers for Extended Negatively Dependent Random Variables

2010 ◽  
Vol 47 (04) ◽  
pp. 908-922 ◽  
Author(s):  
Yiqing Chen ◽  
Anyue Chen ◽  
Kai W. Ng
Keyword(s):  
Law Of Large Numbers ◽  
Large Deviation ◽  
Random Variables ◽  
Renewal Theory ◽  
Independent Random Variables ◽  
Strong Law ◽  
Finite Dimensional ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
New Inequalities

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

scholarly journals Information Transmission under Random Emission Constraints

2014 ◽  
Vol 23 (6) ◽  
pp. 973-1009 ◽  
Author(s):  
FRANCIS COMETS ◽  
FRANÇOIS DELARUE ◽  
RENÉ SCHOTT
Keyword(s):  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Total Duration ◽  
Limited Resources ◽  
Large Numbers ◽  
Coupon Collector ◽  
Selection Of ◽  
Quantities Of Interest

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n → ∞, for the proportion of informed vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton–Watson tree with respect to the success epochs of the coupon collector problem.

scholarly journals DYNAMIC QUANTUM BERNOULLI RANDOM WALKS

2008 ◽  
Vol 11 (02) ◽  
pp. 213-229
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
RENÉ SCHOTT
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Numbers

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

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).

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (03) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Some limit theorems for positive recurrent branching Markov chains: I

1998 ◽  
Vol 30 (03) ◽  
pp. 693-710 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Position Distribution ◽  
Discrete State ◽  
Large Numbers ◽  
Watson Process ◽  
Positive Recurrent ◽  
Discrete State Space

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

Sign in / Sign up

Export Citation Format

Share Document