scholarly journals On the reflected random walk on R+

2017 ◽  
Vol 21 ◽  
pp. 350-368
Author(s):  
Jean−Baptiste Boyer
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Regularity Condition ◽  
Bounded Function ◽  
Markov Operator ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Functional Central Limit ◽  
Reflected Random Walk

Let ρ be a borelian probability measure on R having a moment of order 1 and a drift λ = ∫Rydρ(y) < 0. Consider the random walk on R+ starting at x ∈ R+ and defined for any n ∈N by \begin{eqnarray*} \left\{\begin{array}{rl} X_0&=x \\ X_{n+1} & = |X_n+Y_{n+1}| \end{array}\right. \end{eqnarray*} where (Yn) is an iid sequence of law ρ. We denote P the Markov operator associated to this random walk and, for any borelian bounded function f on R+, we call Poisson’s equation the equation f = g − Pg with unknown function g. In this paper, we prove that under a regularity condition on ρ and f, there is a solution to Poisson’s equation converging to 0 at infinity. Then, we use this result to prove the functional central limit theorem and it’s almost-sure version.

A Martingale Approach to Regenerative Simulation

1995 ◽  
Vol 9 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Peter W. Glynn ◽  
Donald L. Iglehart
Keyword(s):  
Steady State ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Martingale Approach ◽  
Regenerative Simulation

The standard regenerative method for estimating steady-state parameters is extended to permit cycles that begin and end in different states. This result is established using the Dynkin martingale and a related solution to Poisson's equation. We compare the variance constant that appears in the associated central limit theorem with that arising from cycles that begin and end in the same state. The standard regenerative method has a smaller variance constant than does the alternative.

scholarly journals A Combinatorial Approach to a Model of Constrained Random Walkers

2015 ◽  
Vol 25 (2) ◽  
pp. 222-235
Author(s):  
T. ESPINASSE ◽  
N. GUILLOTIN-PLANTARD ◽  
P. NADEAU
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Combinatorial Approach ◽  
Extended Version ◽  
Random Walkers ◽  
Fixed Distance ◽  
Functional Central Limit

In [1], the authors consider a random walk (Zn,1, . . ., Zn,K+1) ∈ ${\mathbb{Z}}$K+1 with the constraint that each coordinate of the walk is at distance one from the following coordinate. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with that obtained in [1].

scholarly journals Central limit theorem for alternating renewal processes

2007 ◽  
Vol 47 ◽  
Author(s):  
Rimas Banys
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Limit Theorems ◽  
Regularity Condition ◽  
Distribution Functions ◽  
Renewal Processes ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

Functional central limit theorems for stationary alternating renewal processes with dependent work and repair times, and for associated workload processes are stated. The weak convergence of distributions of properly scaled processesin the Skorokhodspace holds under some regularity condition imposed on the distribution functions of work and repair times.

scholarly journals A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees

2018 ◽  
Vol 55 (2) ◽  
pp. 610-626 ◽  
Author(s):  
Adam Bowditch
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Upper Bound ◽  
Functional Central Limit Theorem ◽  
Biased Random Walk ◽  
Quenched Central Limit Theorem ◽  
Functional Central Limit

AbstractIn this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

scholarly journals A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

2016 ◽  
Vol 53 (4) ◽  
pp. 1178-1192 ◽  
Author(s):  
Alexander Iksanov ◽  
Zakhar Kabluchko
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Branching Random Walk ◽  
Iterated Logarithm ◽  
Functional Central Limit

Abstract Let (Wn(θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (Wn(2θ))n∈ℕ0 is uniformly integrable and that var W1(θ) is finite, we prove a functional central limit theorem for the tail process (W∞(θ)-Wn+r(θ))r∈ℕ0 and a law of the iterated logarithm for W∞(θ)-Wn(θ) as n→∞.

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