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.

The geometric convergence rate of a Lindley random walk

1997 ◽  
Vol 34 (03) ◽  
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 ‘(r 0) = 0.

scholarly journals Conditional Limit Theorems for the Terms of a Random Walk Revisited

2013 ◽  
Vol 50 (3) ◽  
pp. 871-882
Author(s):  
Shaul K. Bar-Lev ◽  
Ernst Schulte-Geers ◽  
Wolfgang Stadje
Keyword(s):  
Random Walk ◽  
Total Variation ◽  
Gamma Distribution ◽  
Conditional Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Stable Distributions ◽  
Conditional Limit Theorems ◽  
Conditional Limit ◽  
Independent And Identically Distributed

In this paper we derive limit theorems for the conditional distribution ofX1givenSn=snasn→ ∞, where theXiare independent and identically distributed (i.i.d.) random variables,Sn=X1+··· +Xn, andsn/nconverges orsn≡sis constant. We obtain convergence in total variation of PX1∣Sn/n=sto a distribution associated to that ofX1and of PnX1∣Sn=sto a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

Conditional Limit Theorems for the Terms of a Random Walk Revisited

2013 ◽  
Vol 50 (03) ◽  
pp. 871-882
Author(s):  
Shaul K. Bar-Lev ◽  
Ernst Schulte-Geers ◽  
Wolfgang Stadje
Keyword(s):  
Random Walk ◽  
Total Variation ◽  
Gamma Distribution ◽  
Conditional Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Stable Distributions ◽  
Conditional Limit Theorems ◽  
Conditional Limit ◽  
Independent And Identically Distributed

In this paper we derive limit theorems for the conditional distribution of X 1 given S n =s n as n→ ∞, where the X i are independent and identically distributed (i.i.d.) random variables, S n =X 1+··· +X n , and s n /n converges or s n ≡ s is constant. We obtain convergence in total variation of P X 1∣ S n /n=s to a distribution associated to that of X 1 and of P nX 1∣ S n =s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

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, ….)

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 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.

scholarly journals On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold

2009 ◽  
Vol 46 (2) ◽  
pp. 559-570 ◽  
Author(s):  
Xuemiao Ha ◽  
Qihe Tang ◽  
Li Wei
Keyword(s):  
Random Walk ◽  
Asymptotic Formula ◽  
Recent Literature ◽  
Random Variables ◽  
Applied Probability ◽  
Tail Behavior ◽  
Heavy Tailed ◽  
Precise Asymptotic ◽  
Independent And Identically Distributed

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

Poisson recurrence times

1967 ◽  
Vol 4 (3) ◽  
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, Y2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y1 + … + Yn, n = 1,2,…, S0 = 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, ….)

scholarly journals Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

2009 ◽  
Vol 46 (1) ◽  
pp. 255-271 ◽  
Author(s):  
Tasos C. Christofides ◽  
Eutichia Vaggelatou
Keyword(s):  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Partial Sums ◽  
Convex Order ◽  
Absolutely Continuous ◽  
Normal Approximations ◽  
Independent And Identically Distributed

Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

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