Comparing sums of exchangeable Bernoulli random variables

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}.

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On Durbin's Series for the Density of First Passage Times

Journal of Applied Probability ◽

10.1017/s0021900200008263 ◽

2011 ◽

Vol 48 (03) ◽

pp. 713-722

Author(s):

P. Zipkin

Keyword(s):

Pointwise Convergence ◽

First Passage Time ◽

Convergent Series ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Substitution Method ◽

Curved Boundary ◽

Weiner Process ◽

Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

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Weak convergence and first passage times

Journal of Applied Probability ◽

10.1017/s0021900200048014 ◽

1975 ◽

Vol 12 (02) ◽

pp. 324-332

Author(s):

Allan Gut

Keyword(s):

First Passage Time ◽

Domain Of Attraction ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Stable Law ◽

Common Distribution Function ◽

Functional Central Limit Theorems ◽

Functional Central Limit ◽

Passage Times

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp ], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

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On Durbin's Series for the Density of First Passage Times

Journal of Applied Probability ◽

10.1239/jap/1316796909 ◽

2011 ◽

Vol 48 (3) ◽

pp. 713-722

Author(s):

P. Zipkin

Keyword(s):

Pointwise Convergence ◽

First Passage Time ◽

Convergent Series ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Substitution Method ◽

Curved Boundary ◽

Weiner Process ◽

Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

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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.

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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.

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Weak convergence and first passage times

Journal of Applied Probability ◽

10.2307/3212446 ◽

1975 ◽

Vol 12 (2) ◽

pp. 324-332 ◽

Cited By ~ 4

Author(s):

Allan Gut

Keyword(s):

First Passage Time ◽

Domain Of Attraction ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Stable Law ◽

Common Distribution Function ◽

Functional Central Limit Theorems ◽

Functional Central Limit ◽

Passage Times

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

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

Journal of Applied Probability ◽

10.2307/3212200 ◽

1965 ◽

Vol 2 (2) ◽

pp. 377-395 ◽

Cited By ~ 2

Author(s):

J. Chover ◽

G. Yeo

Keyword(s):

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Independent Random Variables ◽

First Passage ◽

Boundary Problems ◽

Finite Sum

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)).

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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.

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

Journal of Applied Probability ◽

10.1017/s002190020010871x ◽

1965 ◽

Vol 2 (02) ◽

pp. 377-395 ◽

Cited By ~ 1

Author(s):

J. Chover ◽

G. Yeo

Keyword(s):

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Independent Random Variables ◽

First Passage ◽

Boundary Problems ◽

Finite Sum

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)).

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