Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (Xn,Sn), n ≥0, in which Xn takes values in a general state space and Sn takes values in ℝd. In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when Sn exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for Sn. For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (XT,ST), where T = inf { n : Sn,1 > b } and Sn,1 denotes the first component of Sn.

Download Full-text

Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

Advances in Applied Probability ◽

10.1017/s0001867800011058 ◽

2001 ◽

Vol 33 (03) ◽

pp. 652-673 ◽

Cited By ~ 2

Author(s):

Cheng-Der Fuh ◽

Tze Leung Lai

Keyword(s):

Random Walks ◽

First Passage Time ◽

Renewal Theory ◽

Passage Time ◽

Boundary Crossing ◽

High Threshold ◽

Renewal Theorem ◽

First Passage ◽

Markov Random ◽

General State Space

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (X n ,S n ), n ≥0, in which X n takes values in a general state space and S n takes values in ℝ d . In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when S n exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for S n . For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (X T ,S T ), where T = inf { n : S n,1 > b } and S n,1 denotes the first component of S n .

Download Full-text

Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

Advances in Applied Probability ◽

10.1239/aap/1189518640 ◽

2007 ◽

Vol 39 (3) ◽

pp. 826-852 ◽

Cited By ~ 1

Author(s):

Cheng-Der Fuh

Keyword(s):

Random Walks ◽

Asymptotic Expansions ◽

First Passage Time ◽

Passage Time ◽

Boundary Crossing ◽

Renewal Theorem ◽

Ladder Height ◽

Markov Random ◽

Markov Random Walk ◽

General State Space

Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Download Full-text

Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

Advances in Applied Probability ◽

10.1017/s0001867800002068 ◽

2007 ◽

Vol 39 (03) ◽

pp. 826-852 ◽

Cited By ~ 1

Author(s):

Cheng-Der Fuh

Keyword(s):

Random Walks ◽

Asymptotic Expansions ◽

First Passage Time ◽

Passage Time ◽

Boundary Crossing ◽

Renewal Theorem ◽

Ladder Height ◽

Markov Random ◽

Markov Random Walk ◽

General State Space

Let {(X n , S n ), n ≥ 0} be a Markov random walk in which X n takes values in a general state space and S n takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Download Full-text

Exact enumeration approach to first-passage time distribution of non-Markov random walks

Physical Review E ◽

10.1103/physreve.99.062101 ◽

2019 ◽

Vol 99 (6) ◽

Cited By ~ 2

Author(s):

Shant Baghram ◽

Farnik Nikakhtar ◽

M. Reza Rahimi Tabar ◽

S. Rahvar ◽

Ravi K. Sheth ◽

...

Keyword(s):

Random Walks ◽

Time Distribution ◽

First Passage Time ◽

Passage Time ◽

Exact Enumeration ◽

First Passage ◽

Markov Random

Download Full-text

Calculations of first passage time of delayed tree-like networks

International Journal of Modern Physics B ◽

10.1142/s0217979215502008 ◽

2015 ◽

Vol 29 (28) ◽

pp. 1550200

Author(s):

Shuai Wang ◽

Weigang Sun ◽

Song Zheng

Keyword(s):

Random Walks ◽

First Passage Time ◽

Passage Time ◽

The Novel ◽

Mean First Passage Time ◽

First Passage ◽

Initial Node ◽

Network Parameters ◽

Self Similar ◽

The Mean

In this paper, we study random walks in a family of delayed tree-like networks controlled by two network parameters, where an immobile trap is located at the initial node. The novel feature of this family of networks is that the existing nodes have a time delay to give birth to new nodes. By the self-similar network structure, we obtain exact solutions of three types of first passage time (FPT) measuring the efficiency of random walks, which includes the mean receiving time (MRT), mean sending time (MST) and mean first passage time (MFPT). The obtained results show that the MRT, MST and MFPT increase with the network parameters. We further show that the values of MRT, MST and MFPT are much shorter than the nondelayed counterpart, implying that the efficiency of random walks in delayed trees is much higher.

Download Full-text