scholarly journals Some results using general moment functions

1969 ◽  
Vol 10 (3-4) ◽  
pp. 429-441 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Random Variables ◽  
Closure Property ◽  
Probability Generating Functions ◽  
Moment Functions ◽  
Result Theorem ◽  
Independent And Identically Distributed

SummaryLet {Xn} be a sequence fo independent and identically distributed random variables such that 0 <μ = εXn ≦ + ∞ and write Sn = X1+X2+ … +Xn. Letv ≧ 0 be an integer and let M(x) be a non-decreasing function of x ≧ 0 such that M(x)/x is non-increasing and M(0) > 0. Then if ε|X1νM(|X1|) < ∞ and μ < ∞ it follows that ε|Sn|νM(|Sn|) ~ (nμ)vM(nμ) as n → ∞. If μ = ∞ (ν = 0) then εM(|Sn|) = 0(n). A variety of results stem from this main theorem (Theorem 2), concerning a closure property of probability generating functions and a random walk result (Theorem 1) connected with queues.

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (01) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Y n , N n ) n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n ) n≥1 defined by Z n = Y n - Σ n-1, where It is shown that the probability that the maximum M = max n≥1 Z n exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

Poisson recurrence times

1967 ◽  
Vol 4 (03) ◽  
pp. 605-608
Author(s):  
Meyer Dwass
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variables ◽  
Recurrent Event ◽  
Recurrence Times ◽  
Image Position ◽  
Independent And Identically Distributed

Let Y1, Y 2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y 1 + … + Yn, n = 1,2,…, S 0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the fn can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2 − 1, ….)

ON THE DISCRETE-TIME G/GI/∞ QUEUE

2008 ◽  
Vol 22 (4) ◽  
pp. 557-585 ◽  
Author(s):  
Iddo Eliazar
Keyword(s):  
Fixed Points ◽  
Discrete Time ◽  
Generating Functions ◽  
Random Variables ◽  
Output Process ◽  
Input Output ◽  
Input Process ◽  
Probability Generating Functions ◽  
Queue Model ◽  
Multidimensional Probability

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The system's output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

scholarly journals Probability generating functions for discrete real-valued random variables

10.4213/tvp8 ◽  
2007 ◽  
Vol 52 (1) ◽  
pp. 129-149
Author(s):  
Manuel Leote Tavares Ingles Esquivel ◽  
Manuel Leote Tavares Ingles Esquivel
Keyword(s):  
Generating Functions ◽  
Random Variables ◽  
Probability Generating Functions

A random walk problem with correlation

1972 ◽  
Vol 9 (02) ◽  
pp. 436-440 ◽  
Author(s):  
A. D. Proudfoot ◽  
D. G. Lampard
Keyword(s):  
Random Walk ◽  
Green’S Function ◽  
Green's Function ◽  
Generating Functions ◽  
Transition Probability ◽  
Higher Order ◽  
Order Transition ◽  
Probability Generating Functions ◽  
Discrete Domain

The higher-order transition probability generating functions for a random-walk with correlation between steps is calculated as a discrete-domain Green's function.

Sooner and later waiting time problems for patterns in Markov dependent trials

2003 ◽  
Vol 40 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Qing Han ◽  
Katuomi Hirano
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Random Variables ◽  
Probability Functions ◽  
Probability Generating Functions ◽  
Markov Dependent Trials

In this paper, we investigate sooner and later waiting time problems for patterns S0 and S1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of S0 and between successive occurrences of S0 and S1 and of the waiting time until the rth occurrence of S0 are also given.

The geometric convergence rate of a Lindley random walk

1997 ◽  
Vol 34 (3) ◽  
pp. 806-811
Author(s):  
Robert B. Lund
Keyword(s):  
Random Walk ◽  
Convergence Rate ◽  
Large Deviations ◽  
Total Variation ◽  
Short Proof ◽  
Random Variables ◽  
Invariant Distribution ◽  
Initial State ◽  
Geometric Convergence ◽  
Independent And Identically Distributed

Let {Xn} be the Lindley random walk on [0,∞) defined by . Here, {An} is a sequence of independent and identically distributed random variables. When for some r > 1, {Xn} converges at a geometric rate in total variation to an invariant distribution π; i.e. there exists s > 1 such that for every initial state . In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels (1975, 1976): the largest s satisfying the above relationship is and satisfies φ ‘(r0) = 0.

scholarly journals On perpetuities with gamma-like tails

2018 ◽  
Vol 55 (2) ◽  
pp. 368-389 ◽  
Author(s):  
Dariusz Buraczewski ◽  
Piotr Dyszewski ◽  
Alexander Iksanov ◽  
Alexander Marynych
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Exponential Moments ◽  
The One ◽  
The Right ◽  
Independent And Identically Distributed

Abstract An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We provide three disjoint groups of sufficient conditions which ensure that the right tail of a perpetuity ℙ{X > x} is asymptotic to axce-bx as x → ∞ for some a, b > 0, and c ∈ ℝ. Our results complement those of Denisov and Zwart (2007). As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. We give several examples in which the distributions of perpetuities are explicitly identified.

Branching and clustering models associated with the ‘lost-games distribution’

1969 ◽  
Vol 6 (3) ◽  
pp. 700-703 ◽  
Author(s):  
Adrienne W. Kemp ◽  
C. D. Kemp
Keyword(s):  
Generating Functions ◽  
Random Variables ◽  
Probability Generating Functions ◽  
Generalized Distributions

We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variables Xi with probability generating functions gi(s), i = 1, 2, 3, if g3(s) = g1[g2(s)], we say that X3 is X1 generalized by X2 and write X3~ X1VX2.

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