Solutions of some two-sided boundary problems for sums of variables with alternating distributions

In this paper we present a method for obtaining explicit results for some two-sided boundary problems involving sums of independent random variables with alternating distributions. We apply the method to finding the first passage time to either one of two finite barriers, and to some situations arising in queueing and dam theory. The results can be expressed in terms of a finite sum of simple repeated integrals (or sums) of known functions (cf. formulae (3.6)– (3.11)).

Download Full-text

Comparing sums of exchangeable Bernoulli random variables

Journal of Applied Probability ◽

10.1017/s0021900200099733 ◽

1996 ◽

Vol 33 (02) ◽

pp. 285-310 ◽

Cited By ~ 5

Author(s):

Claude Lefèvre ◽

Sergey Utev

Keyword(s):

Iterative Procedure ◽

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Sampling Schemes ◽

Bernoulli Random Variables ◽

Discrete Random Variables ◽

Comparison Results

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

Download Full-text

Comparing sums of exchangeable Bernoulli random variables

Journal of Applied Probability ◽

10.2307/3215055 ◽

1996 ◽

Vol 33 (2) ◽

pp. 285-310 ◽

Cited By ~ 32

Author(s):

Claude Lefèvre ◽

Sergey Utev

Keyword(s):

Iterative Procedure ◽

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Sampling Schemes ◽

Bernoulli Random Variables ◽

Discrete Random Variables ◽

Comparison Results

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

Download Full-text

A limit theorem for random walks with drift

Journal of Applied Probability ◽

10.2307/3212307 ◽

1967 ◽

Vol 4 (1) ◽

pp. 144-150 ◽

Cited By ~ 11

Author(s):

C. C. Heyde

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Random Walks ◽

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Random Walk Process ◽

First Passage ◽

Time Out ◽

Image Position

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

Download Full-text

A limit theorem for random walks with drift

Journal of Applied Probability ◽

10.1017/s0021900200025304 ◽

1967 ◽

Vol 4 (01) ◽

pp. 144-150 ◽

Cited By ~ 3

Author(s):

C. C. Heyde

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Random Walks ◽

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Random Walk Process ◽

First Passage ◽

Time Out ◽

Image Position

Let Xi , i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

Download Full-text

Invariance Theorems for First Passage Time Random Variables

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-031-9 ◽

1972 ◽

Vol 15 (2) ◽

pp. 171-176 ◽

Cited By ~ 6

Author(s):

A. K. Basu

Keyword(s):

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Wiener Measure ◽

First Passage ◽

Image Position

Let X1X2,… be i.i.d. r.v. with EX=μ>0, and E(X-μ)2 = σ2<∞.Let Sk=X1+…+Xk and vx=max{k:Sk≤x}, x≥0 and vx=0 if X1>x. Billingsley [1] proved if X1≥0 thenconverges weakly to the Wiener measure W.Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove thatconverges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.

Download Full-text

First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

Download Full-text