The nonhomogeneous frog model on ℤ

2018 ◽  
Vol 55 (4) ◽  
pp. 1093-1112
Author(s):  
Josh Rosenberg
Keyword(s):  
Positive Integer ◽  
Random Walks ◽  
Stochastic Order ◽  
Random Variables ◽  
Independent Random Variables ◽  
Active Particle ◽  
Integer Points ◽  
Interacting Random Walks ◽  
Frog Model ◽  
Positive Integers

Abstract We examine a system of interacting random walks with leftward drift on ℤ, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point possess equal leftward drift. Once activated, the trajectories of distinct particles are independent. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Additional conditions that we impose on our model include that the number of frogs (i.e. particles) at positive integer points is a sequence of independent random variables which is increasing in terms of the standard stochastic order, and that the sequence of leftward drifts associated with frogs originating at these points is decreasing. Our results include sharp conditions with respect to the sequence of random variables and the sequence of drifts that determine whether the model is transient (meaning the probability infinitely many frogs return to the origin is 0) or nontransient. We consider several, more specific, versions of the model described, and a cleaner, more simplified set of sharp conditions will be established for each case.

Some renewal theorems for random walks in multidimensional time

1984 ◽  
Vol 95 (1) ◽  
pp. 149-154 ◽  
Author(s):  
Makoto Maejima ◽  
Toshio Mori
Keyword(s):  
Random Walks ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sum ◽  
Positive Integers ◽  
Independent Identically Distributed

Let Kr denote the set of r-tuples n = (n1n2, …, nr) with positive integers as coordinates (r ≥ 1) and {X, Xn, n ε Kr} be a family of independent, identically distributed random variables with positive mean 0 < EX ≡ μ < ∞ and finite positive variance 0 < var X ≡ σ2 ∞. The notation m ≤ n, where m = (mi) and n = (ni), means that mi ≤ ni (i = 1, 2,…, r) and |n| = n1n2 … nr. Denote Sn =Σj ≤ nXj (j, n ε Kr). When r = 1, {Xn, n ε Kr) reduces to the sequence {Xj, j ε 1} of independent random variables each distributed as X, and Sn becomes the ordinary partial sum .

scholarly journals Frogs and some other interacting random walks models

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Serguei Yu. Popov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Active Particle ◽  
Open Problems ◽  
Simple Version ◽  
Random Walks On Graphs ◽  
Interacting Random Walks ◽  
International Audience ◽  
Frog Model

International audience We review some recent results for a system of simple random walks on graphs, known as \emphfrog model. Also, we discuss several modifications of this model, and present a few open problems. A simple version of the frog model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active.

scholarly journals Empirical processes for recurrent and transient random walks in random scenery

2020 ◽  
Vol 24 ◽  
pp. 127-137
Author(s):  
Nadine Guillotin-Plantard ◽  
Françoise Pène ◽  
Martin Wendler
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Random Walks ◽  
Stable Distribution ◽  
Random Variables ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Random Scenery ◽  
Recurrent Random Walk ◽  
Transient Random Walk

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

Stochastic comparisons of random minima and maxima

1997 ◽  
Vol 34 (02) ◽  
pp. 420-425 ◽  
Author(s):  
Moshe Shaked ◽  
Tityik Wong
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Stochastic Comparison ◽  
Stochastic Comparisons ◽  
Comparison Results

Let X 1, X 2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X 1, X 2,…, XN ) and max{X 1, X 2,…, XN }.

scholarly journals Deviation bounds for the first passage time in the frog model

2019 ◽  
Vol 51 (01) ◽  
pp. 184-208 ◽  
Author(s):  
Naoki Kubota
Keyword(s):  
Random Walks ◽  
Large Deviation ◽  
First Passage Time ◽  
Passage Time ◽  
The Other ◽  
Active Particle ◽  
First Passage ◽  
Cubic Lattice ◽  
Concentration Bounds ◽  
Frog Model

AbstractWe consider the so-called frog model with random initial configurations. The dynamics of this model are described as follows. Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

On the asymptotic independent representations for sums of some weakly dependent random variables

2006 ◽  
Vol 43 (1) ◽  
pp. 33-46
Author(s):  
Rafik Aguech ◽  
Sana Louhichi ◽  
Sofyen Louhichi
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Triangular Array ◽  
Independent Random Variables ◽  
Dependent Random Variables ◽  
Limiting Behavior ◽  
Weakly Dependent Random Variables ◽  
Weakly Dependent

Let, for each n?N, (Xi,n)0?i?nbe a triangular array of stationary, centered, square integrable and associated real valued random variables satisfying the weakly dependence condition lim N?N0limsup n?+8nSr=NnCov (X0,n, Xr,n)=0;where N0is either infinite or the first positive integer Nfor which the limit of the sum nSr=NnCov (X0,n, Xr,n) vanishes as n goes to infinity. The purpose of this paper is to build, from (Xi,n)0?i?n, a sequence of independent random variables (X˜i,n)0?i?nsuch that the two sumsSi=1nXi,nandSi=1nX˜i,nhave the same asymptotic limiting behavior (in distribution).

On the range of the transient frog model on ℤ

2017 ◽  
Vol 49 (2) ◽  
pp. 327-343 ◽  
Author(s):  
Arka Ghosh ◽  
Steven Noren ◽  
Alexander Roitershtein
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Explicit Formula ◽  
Infinite System ◽  
Scaling Limits ◽  
Interacting Random Walks ◽  
Degenerate Distribution ◽  
Frog Model ◽  
Asymptotic Scaling

Abstract We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

A note on random walks in multidimensional time

1986 ◽  
Vol 99 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Janos Galambos ◽  
Imre Kátai
Keyword(s):  
Random Walks ◽  
Recent Work ◽  
Random Variables ◽  
Finite Variance ◽  
Image Position ◽  
Positive Integers ◽  
Independent And Identically Distributed

Let Kr denote the set of r-tuples n = (n1, n2, …, nr), r ≥ 1, where the components ni are positive integers. Let {X, Xn, n ∈ Kr} be a family of independent and identically distributed random variables with positive mean EX = μ < + ∞ and finite variance VX = σ2 < + ∞. In a recent work, M. Maejima and T. Mori [2] have shown that, if X is integer valued, aperiodic and E∣X∣3 < + ∞, then, for r = 2 or 3,wherethe summation being extended over all members j = (j1,j2, …, jr) of Kr that satisfy jt ≤ nt for all 1 ≤ t ≤ r.

scholarly journals Mean convergence theorems and weak laws of large numbers for double arrays of random variables

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Le Van Thanh
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Convergence Theorems ◽  
Laws Of Large Numbers ◽  
Mean Convergence ◽  
Convergence Results ◽  
Large Numbers ◽  
The Mean ◽  
Weak Laws ◽  
Positive Integers

For a double array of random variables {Xmn, m ≥ 1, n ≥ 1}, mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which ∑i=1km∑j=1lnamnij(Xij−EXij)→Lr0(0<r≤2) where {amnij;m,n,i,j≥1} are constants, and {kn,n≥1} and {ln,n≥1} are sequences of positive integers. The weak law results provide conditions for ∑i=1Tm∑j=1τnamnij(Xij−EXij)→p0 to hold where {Tm,m≥1} and {τn,n≥1} are sequences of positive integer-valued random variables. The sharpness of the results is illustrated by examples.

scholarly journals On trigonometric sums with random frequencies

2018 ◽  
Vol 55 (1) ◽  
pp. 141-152
Author(s):  
Alina Bazarova ◽  
István Berkes ◽  
Marko Raseta
Keyword(s):  
Uniform Distribution ◽  
Lebesgue Measure ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Independent Random Variables ◽  
Trigonometric Sums ◽  
Positive Axis ◽  
Positive Integers

We prove that if Ik are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω,F,P) such that nk is uniformly distributed on Ik, then has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),B, λ), where B is the Borel σ-algebra and λ is the Lebesgue measure. We also investigate the case when nk have continuous uniform distribution on disjoint intervals Ik on the positive axis.

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