Random walks with negative drift conditioned to stay positive

1974 ◽  
Vol 11 (4) ◽  
pp. 742-751 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Principal Result ◽  
Collective Risk ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Independent Identically Distributed

Let {Xk: k ≧ 1} be a sequence of independent, identically distributed random variables with EX1 = μ < 0. Form the random walk {Sn: n ≧ 0} by setting S0 = 0, Sn = X1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X1) that Sn, conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Random walks with negative drift conditioned to stay positive

1974 ◽  
Vol 11 (04) ◽  
pp. 742-751 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Principal Result ◽  
Collective Risk ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Independent Identically Distributed

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ &lt; 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X 1) that Sn , conditioned on T &gt; n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ &lt; 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n &lt; N &lt; ∞ converge to those of the Brownian excursion process.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (2) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξk : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ1} = μ ≠ 0. Form the random walk {Sn : n ≧ 0} by setting S0, Sn = ξ1 + ξ2 + ··· + ξn, n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ1) that the finite-dimensional distributions of Xn, conditioned on n < N < ∞ converge to those of the Brownian excursion process.

Series expansions for the all-time maximum of α-stable random walks

2016 ◽  
Vol 48 (3) ◽  
pp. 744-767
Author(s):  
Clifford Hurvich ◽  
Josh Reed
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Series Expansions ◽  
Stable Random Variables ◽  
The Mean ◽  
The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

scholarly journals Some approximate results in renewal and dam theories

1971 ◽  
Vol 12 (4) ◽  
pp. 425-432 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walk ◽  
Sequential Analysis ◽  
Random Variables ◽  
Renewal Theory ◽  
Slight Modification ◽  
Exact Results ◽  
Fundamental Identity ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Two Barriers

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

Dirichlet Random Walks

2014 ◽  
Vol 51 (04) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk inRd. We compute explicitly, for a number of cases, the distribution of the random variableWusing a form of Stieltjes transform ofWinstead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere ofRd.

Subsequence principles for vector-valued random variables

1979 ◽  
Vol 86 (2) ◽  
pp. 301-312 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Principal Result ◽  
Moment Condition ◽  
Special Cases ◽  
Cesaro Averages ◽  
Image Position ◽  
Vector Valued ◽  
Subsequence Principle ◽  
Independent Identically Distributed

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such thatalmost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

On the number of jumps of random walks with a barrier

2008 ◽  
Vol 40 (01) ◽  
pp. 206-228 ◽  
Author(s):  
Alex Iksanov ◽  
Martin Möhle
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Tail Behavior ◽  
Limiting Distributions ◽  
Current State ◽  
Single Block ◽  
The Right ◽  
First Time

LetS0:= 0 andSk:=ξ1+ ··· +ξkfork∈ ℕ := {1, 2, …}, where {ξk:k∈ ℕ} are independent copies of a random variableξwith values in ℕ and distributionpk:= P{ξ=k},k∈ ℕ. We interpret the random walk {Sk:k= 0, 1, 2, …} as a particle jumping to the right through integer positions. Fixn∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal ton. This constraint defines an increasing Markov chain {Rk(n):k= 0, 1, 2, …} which never reaches the staten. We call this process a random walk with barriern. LetMndenote the number of jumps of the random walk with barriern. This paper focuses on the asymptotics ofMnasntends to ∞. A key observation is that, underp1&gt; 0, {Mn:n∈ ℕ} satisfies the distributional recursionM1= 0 andforn= 2, 3, …, whereInis independent ofM2, …,Mn−1with distribution P{In=k} =pk/ (p1+ ··· +pn−1),k∈ {1, …,n− 1}. Depending on the tail behavior of the distribution ofξ, several scalings forMnand corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps,Mn, with the first time,Nn, when the unrestricted random walk {Sk:k= 0, 1, …} reaches a state larger than or equal ton. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) forβ(a,b)-coalescent processes with parameters 0 &lt;a&lt; 2 andb= 1.

scholarly journals On a class of random walks in simplexes

2020 ◽  
Vol 57 (2) ◽  
pp. 409-428
Author(s):  
Tuan-Minh Nguyen ◽  
Stanislav Volkov
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Beta Distribution ◽  
Interior Point ◽  
Random Variable ◽  
Limiting Distributions ◽  
Special Cases ◽  
Limit Behaviour ◽  
New Location

AbstractWe study the limit behaviour of a class of random walk models taking values in the standard d-dimensional ( $d\ge 1$ ) simplex. From an interior point z, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on z, and then the particle randomly jumps to a new location z′ on the segment connecting z to the chosen vertex. In some special cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are Dirichlet. We also consider a related history-dependent random walk model in [0, 1] based on an urn-type scheme. We show that this random walk converges in distribution to an arcsine random variable.

scholarly journals Scaling exponents of step-reinforced random walks

2020 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Scaling Exponent ◽  
Basic Algorithm ◽  
Real Random Variable ◽  
Scaling Exponents ◽  
Reinforced Random Walks ◽  
Speed Up ◽  
With Memory

Abstract Let $$X_1, X_2, \ldots $$ X 1 , X 2 , … be i.i.d. copies of some real random variable X. For any deterministic $$\varepsilon _2, \varepsilon _3, \ldots $$ ε 2 , ε 3 , … in $$\{0,1\}$$ { 0 , 1 } , a basic algorithm introduced by H.A. Simon yields a reinforced sequence $$\hat{X}_1, \hat{X}_2 , \ldots $$ X ^ 1 , X ^ 2 , … as follows. If $$\varepsilon _n=0$$ ε n = 0 , then $$ \hat{X}_n$$ X ^ n is a uniform random sample from $$\hat{X}_1, \ldots , \hat{X}_{n-1}$$ X ^ 1 , … , X ^ n - 1 ; otherwise $$ \hat{X}_n$$ X ^ n is a new independent copy of X. The purpose of this work is to compare the scaling exponent of the usual random walk $$S(n)=X_1+\cdots + X_n$$ S ( n ) = X 1 + ⋯ + X n with that of its step reinforced version $$\hat{S}(n)=\hat{X}_1+\cdots + \hat{X}_n$$ S ^ ( n ) = X ^ 1 + ⋯ + X ^ n . Depending on the tail of X and on asymptotic behavior of the sequence $$(\varepsilon _n)$$ ( ε n ) , we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.

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