Random Walk, Brownian Motion, and the Strong Markov Property

2021 ◽  
pp. 71-97
Author(s):  
Rabi Bhattacharya ◽  
Edward C. Waymire
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Markov Property ◽  
Strong Markov Property ◽  
Strong Markov

scholarly journals Asymptotics for the moments of the overshoot and undershoot of a random walk

2009 ◽  
Vol 41 (2) ◽  
pp. 469-494 ◽  
Author(s):  
Zhaolei Cui ◽  
Yuebao Wang ◽  
Kaiyong Wang
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Strong Markov Property ◽  
Renewal Equations ◽  
Ladder Height ◽  
Overshoot And Undershoot ◽  
Equivalent Conditions ◽  
Strong Markov

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

scholarly journals Asymptotics for the moments of the overshoot and undershoot of a random walk

2009 ◽  
Vol 41 (02) ◽  
pp. 469-494
Author(s):  
Zhaolei Cui ◽  
Yuebao Wang ◽  
Kaiyong Wang
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Strong Markov Property ◽  
Renewal Equations ◽  
Ladder Height ◽  
Overshoot And Undershoot ◽  
Equivalent Conditions ◽  
Strong Markov

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

scholarly journals Homogeneity and the Strong Markov Property

1987 ◽  
Vol 15 (1) ◽  
pp. 213-240 ◽  
Author(s):  
Olav Kallenberg
Keyword(s):  
Markov Property ◽  
Strong Markov Property ◽  
Strong Markov
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