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.

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.

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.

Spitzer's condition for asymptotically symmetric random walk

1980 ◽  
Vol 17 (03) ◽  
pp. 856-859
Author(s):  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Length Distribution ◽  
Domain Of Attraction ◽  
Step Length ◽  
Sufficient Condition ◽  
Symmetric Form ◽  
Stable Law ◽  
Symmetric Random Walk ◽  
Image Position

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that lim x→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.

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.

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