scholarly journals On the ladder heights of random walks attracted to stable laws of exponent 1

2018 ◽  
Vol 23 ◽  
Author(s):  
Kôhei Uchiyama
Keyword(s):  
Random Walks ◽  
Stable Laws ◽  
Ladder Heights

Moments of ladder heights in random walks

1980 ◽  
Vol 17 (1) ◽  
pp. 248-252 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Random Walks ◽  
Step Length ◽  
Ladder Heights

A well-known result in the theory of random walks states that E{X2} is finite if and only if E{Z+} and E{Z_} are both finite (Z+ and Z_ being the ladder heights and X a typical step-length) in which case E{X2} = 2E{Z+}E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z+ and Z_ of order ß – 1. The main result is that if β > 2 E{|X|β} < ∞ if and only if and are both finite.

On overshoots and hitting times for random walks

1999 ◽  
Vol 36 (2) ◽  
pp. 593-600
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Hitting Time ◽  
Hitting Times ◽  
First Hitting Time ◽  
Fixed Integer ◽  
Ladder Heights ◽  
Regenerative Sets

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

scholarly journals Stable laws and spectral gap properties for affine random walks

2015 ◽  
Vol 51 (1) ◽  
pp. 319-348
Author(s):  
Zhiqiang Gao ◽  
Yves Guivarc’h ◽  
Émile Le Page
Keyword(s):  
Random Walks ◽  
Spectral Gap ◽  
Stable Laws

On overshoots and hitting times for random walks

1999 ◽  
Vol 36 (02) ◽  
pp. 593-600
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Hitting Time ◽  
Hitting Times ◽  
First Hitting Time ◽  
Fixed Integer ◽  
Ladder Heights ◽  
Regenerative Sets

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x &gt; 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

Moments of ladder heights in random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 248-252 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Random Walks ◽  
Step Length ◽  
Ladder Heights

A well-known result in the theory of random walks states that E{X 2} is finite if and only if E{Z+ } and E{Z_} are both finite (Z + and Z_ being the ladder heights and X a typical step-length) in which case E{X 2} = 2E{Z+ }E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z + and Z_ of order ß – 1. The main result is that if β &gt; 2 E{|X|β} &lt; ∞ if and only if and are both finite.

Non-homogeneous Random Walks

2016 ◽  
Author(s):  
Mikhail Menshikov ◽  
Serguei Popov ◽  
Andrew Wade
Keyword(s):  
Random Walks
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