The stochastic geyser problem for first-passage times

1981 ◽  
Vol 18 (01) ◽  
pp. 91-103
Author(s):  
Josef Steinebach
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
First Passage Times ◽  
First Passage ◽  
Single Realization ◽  
Independent Identically Distributed ◽  
Passage Times

Let X 1, X 2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn 's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).

The stochastic geyser problem for first-passage times

1981 ◽  
Vol 18 (1) ◽  
pp. 91-103 ◽  
Author(s):  
Josef Steinebach
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
First Passage Times ◽  
First Passage ◽  
Single Realization ◽  
Independent Identically Distributed ◽  
Passage Times

Let X1, X2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).

Inequalities for Means of Restricted First-Passage Times in Percolation Theory

1999 ◽  
Vol 8 (4) ◽  
pp. 307-315 ◽  
Author(s):  
SVEN ERICK ALM ◽  
JOHN C. WIERMAN
Keyword(s):  
Half Space ◽  
Percolation Theory ◽  
First Passage Times ◽  
First Passage ◽  
Geometric Argument ◽  
Convexity And Concavity ◽  
Passage Times

A simple geometric argument establishes an inequality between the sums of two pairs of first-passage times. This result is used to prove monotonicity, convexity and concavity results for first-passage times with cylinder and half-space restrictions.

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

Sign in / Sign up

Export Citation Format

Share Document