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.

Some limit theorems for positive recurrent branching Markov chains: I

1998 ◽  
Vol 30 (3) ◽  
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.

Some limit theorems for positive recurrent branching Markov chains: II

1998 ◽  
Vol 30 (03) ◽  
pp. 711-722 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
General State ◽  
Position Distribution ◽  
Large Numbers ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Watson Process ◽  
Positive Recurrent

In this paper we consider a Galton-Watson process in which particles move according to a positive recurrent Markov chain on a general state space. We prove a law of large numbers for the empirical position distribution and also discuss the rate of this convergence.

Some limit theorems for positive recurrent branching Markov chains: II

1998 ◽  
Vol 30 (3) ◽  
pp. 711-722 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
General State ◽  
Position Distribution ◽  
Large Numbers ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Watson Process ◽  
Positive Recurrent

In this paper we consider a Galton-Watson process in which particles move according to a positive recurrent Markov chain on a general state space. We prove a law of large numbers for the empirical position distribution and also discuss the rate of this convergence.

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 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.

Stochastic scrabble: large deviations for sequences with scores

1988 ◽  
Vol 25 (1) ◽  
pp. 106-119 ◽  
Author(s):  
Richard Arratia ◽  
Pricilla Morris ◽  
Michael S. Waterman
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Additive Functional ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Similar Treatment ◽  
Large Numbers ◽  
Large Deviation Result

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

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.

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.

Stochastic scrabble: large deviations for sequences with scores

1988 ◽  
Vol 25 (01) ◽  
pp. 106-119
Author(s):  
Richard Arratia ◽  
Pricilla Morris ◽  
Michael S. Waterman
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Additive Functional ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Similar Treatment ◽  
Large Numbers ◽  
Large Deviation Result

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i)) i∊S letters from a finite alphabet S and v=(v(i)) i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i 1 · ·· im is defined to be V(w)=v(i 1) + · ·· + v(im ). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X 1 X 2 · ·· Xn and Y 1 Y 2 · ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q , v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included. Implicit in these results is a large-deviation result for the additive functional, H ≡ Σ n &lt; τ v(Xn ), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanqing Wang ◽  
Quansheng Liu
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Moderate Deviation ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Branching Process With Immigration

Abstract This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 p\geq 1 , and the boundedness of the harmonic moments E ⁢ W n - a \mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log ⁡ Z n \log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.

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