Further limit theorems for the range of random walk

1974 ◽  
Vol 27 (1) ◽  
pp. 94-117 ◽  
Author(s):  
Naresh C. Jain ◽  
William E. Pruitt
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Range Of Random Walk

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

2000 ◽  
Vol 32 (01) ◽  
pp. 159-176 ◽  
Author(s):  
Markus Bachmann
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Number ◽  
Time Lag ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Branching Random Walk ◽  
Minimal Position ◽  
Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (01) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain,obtained by restarting the original chain at a fixed state after each absorption. The limiting age,A(j), is the weak limit of the timegivenXn=j(n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems forA(J) (J →∞) are given for these examples.

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (1) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

scholarly journals Limit theorems for a minimal random walk model

2019 ◽  
Vol 2019 (8) ◽  
pp. 083206 ◽  
Author(s):  
Cristian F Coletti ◽  
Renato J Gava ◽  
Lucas R de Lima
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Walk Model

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (01) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

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