On the number of jumps of random walks with a barrier

Abstract

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> 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 <a< 2 andb= 1.