Derivatives of the first level-crossing time distribution

2021 ◽  
pp. 271-290
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Time Distribution ◽  
Level Crossing ◽  
Crossing Time ◽  
Derivatives Of

scholarly journals First-crossing and ballot-type results for some nonstationary sequences

2007 ◽  
Vol 39 (02) ◽  
pp. 492-509 ◽  
Author(s):  
Claude Lefèvre
Keyword(s):  
Time Distribution ◽  
Numerical Evaluation ◽  
Random Variables ◽  
Parametric Family ◽  
Partial Sums ◽  
Variable Parameters ◽  
Type Formula ◽  
Polynomial Structure ◽  
Crossing Time ◽  
First Crossing Time

In this paper we consider the problem of first-crossing from above for a partial sums process m+S t , t ≥ 1, with the diagonal line when the random variables X t , t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the X t s belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

Level-crossing time statistics of Gaussian 1/fαnoises

2003 ◽  
Author(s):  
Robert Mingesz ◽  
Zoltan Gingl ◽  
Peter Makra
Keyword(s):  
Level Crossing ◽  
Crossing Time

scholarly journals First-crossing and ballot-type results for some nonstationary sequences

2007 ◽  
Vol 39 (2) ◽  
pp. 492-509 ◽  
Author(s):  
Claude Lefèvre
Keyword(s):  
Time Distribution ◽  
Numerical Evaluation ◽  
Random Variables ◽  
Parametric Family ◽  
Partial Sums ◽  
Variable Parameters ◽  
Type Formula ◽  
Polynomial Structure ◽  
Crossing Time ◽  
First Crossing Time

In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

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