Intersection of independent regenerative sets

1999 ◽  
Vol 114 (1) ◽  
pp. 97-121 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Regenerative Sets

scholarly journals Representations of hermite processes using local time of intersecting stationary stable regenerative sets

2020 ◽  
Vol 57 (4) ◽  
pp. 1234-1251
Author(s):  
Shuyang Bai
Keyword(s):  
Long Range ◽  
Local Time ◽  
Limit Theorems ◽  
Local Times ◽  
Long Range Dependence ◽  
Random Interval ◽  
Stationary Increments ◽  
Self Similar ◽  
Regenerative Sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

Regenerative sets and subordinators

1977 ◽  
Vol 14 (1) ◽  
pp. 217-220
Author(s):  
Cristina Gzyl
Keyword(s):  
Markov Processes ◽  
Probabilistic Structure ◽  
Regenerative Sets

It has been shown that the probabilistic structure of a regular regenerative set can be described in terms of the exponent of an appropriate subordinator. We give a relation which expresses the distribution of two Markov processes associated with a regenerative set in terms of this exponent.

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.

Regenerative sets and subordinators

1977 ◽  
Vol 14 (01) ◽  
pp. 217-220
Author(s):  
Cristina Gzyl
Keyword(s):  
Markov Processes ◽  
Probabilistic Structure ◽  
Regenerative Sets

It has been shown that the probabilistic structure of a regular regenerative set can be described in terms of the exponent of an appropriate subordinator. We give a relation which expresses the distribution of two Markov processes associated with a regenerative set in terms of this exponent.

Topics on regenerative sets and subordinators

1980 ◽  
Vol 12 (02) ◽  
pp. 273-274
Author(s):  
B. Maisonneuve
Keyword(s):  
Regenerative Sets

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 > 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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