Stationary Sequences in a Random Time Scenery

2019 ◽  
pp. 319-344
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Random Walk ◽  
Random Time ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Invariance Principles ◽  
Weak Invariance ◽  
Functional Clt ◽  
Random Scenery

In this chapter, we analyze the asymptotic behavior of the partial sums process associated with examples of stationary sequences in a random time scenery. The examples considered are stationary sequences sampled by shifted renewal Markov chains and random walks in a strictly stationary scenery. The asymptotic behavior of the partial sums process is essentially investigated with the help of the weak invariance principles stated in Chapter 4. More precisely, for the partial sums process associated with a stationary process sampled by a renewal Markov chain stated at zero, due to the non-stationarity of the underlying sequence, the functional CLT is obtained as an application of the functional CLT for non-stationary sequences developed in Section 4.4. In the case where we are sampling a strictly stationary random scenery by a random walk, stationarity is preserved, and the invariance principle is then derived by using the functional CLT under Maxwell–Woodroofe condition.

Random Walk in Random Scenery

2019 ◽  
pp. 382-404
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Random Walk ◽  
Central Limit ◽  
Functional Form ◽  
Stationary Sequence ◽  
Limit Behavior ◽  
Partial Sums ◽  
Determinantal Process ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Random Scenery

In this chapter we investigate the question of central limit behavior and its functional form for the partial sums associated with a centered L2-stationary sequence of real-valued random variables (usually called the random scenery) sampled by a recurrent one-dimensional strongly aperiodic random walk. This question is handled under various conditions dependent on the random scenery. In particular, we assume that the random scenery either satisfies an asymptotic negative dependence condition, or is a function of a determinantal process and a Gaussian sequence, or satisfies a mild projective criterion. We first show that study of central limit behavior for such random walks in random scenery can be handled with results related to linear statistics developed in Chapter 12, provided the random walk has good properties. We then look extensively at the properties of a recurrent one-dimensional strongly aperiodic random walk. The functional form of the central limit theorem is also investigated.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (X n ) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

On the Distribution of the Number of Zero Partial Sums

1969 ◽  
Vol 6 (01) ◽  
pp. 162-176 ◽  
Author(s):  
Ora Engelberg Percus ◽  
Jerome K. Percus
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Generating Function ◽  
General Distribution ◽  
Partial Sums ◽  
Positive Direction ◽  
Negative Direction

A weighted Markov chain technique is used to determine the general distribution of the number of tie positions in a two candidate ballot. This may be interpreted as the distribution of the number of returns to the origin in an asymmetric random walk, when the particle at each step moves μ units in the positive direction and γ units in the negative direction, given the total number of positive steps and the number of negative steps. Explicit results are obtained when γ = 1, whereas the case of μ and γ ≠ 1 is solved in the form of a generating function.

First-passage times for the partial sums of a sequence of geometric distributions

1982 ◽  
Vol 19 (02) ◽  
pp. 430-432
Author(s):  
A. J. Woods
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
First Passage Times ◽  
Partial Sums ◽  
Homogeneous Markov Chain ◽  
First Passage ◽  
Epidemic Process ◽  
Passage Times

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

Reversible Markov Chains

2019 ◽  
pp. 405-427
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Gaussian Approximation ◽  
Building Blocks ◽  
Partial Sums ◽  
Reversible Markov Chain ◽  
Maximal Inequalities ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.

On the Distribution of the Number of Zero Partial Sums

1969 ◽  
Vol 6 (1) ◽  
pp. 162-176 ◽  
Author(s):  
Ora Engelberg Percus ◽  
Jerome K. Percus
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Generating Function ◽  
General Distribution ◽  
Partial Sums ◽  
Positive Direction ◽  
Negative Direction

A weighted Markov chain technique is used to determine the general distribution of the number of tie positions in a two candidate ballot. This may be interpreted as the distribution of the number of returns to the origin in an asymmetric random walk, when the particle at each step moves μ units in the positive direction and γ units in the negative direction, given the total number of positive steps and the number of negative steps. Explicit results are obtained when γ = 1, whereas the case of μ and γ ≠ 1 is solved in the form of a generating function.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (1) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (Xn) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Yn. Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

First-passage times for the partial sums of a sequence of geometric distributions

1982 ◽  
Vol 19 (2) ◽  
pp. 430-432 ◽  
Author(s):  
A. J. Woods
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
First Passage Times ◽  
Partial Sums ◽  
Homogeneous Markov Chain ◽  
First Passage ◽  
Epidemic Process ◽  
Passage Times

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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