## On the number of jumps of random walks with a barrier.(English)Zbl 1157.60041

Let $$S_0= 0$$ and $$S_k= \xi_1+\cdots +\xi_k$$, $$k\in\mathbb{N}$$, where $$\{\xi_k: k\in\mathbb{N}\}$$ is a sequence of i.i.d. $$\mathbb{N}$$-valued random variables, fix $$n\in\mathbb{N}$$, and define a random walk with barrier $$n$$, $$\{R^{(n)}_k: k\in\{0\}\cup\mathbb{N}\}$$, by $$R^{(n)}_0= 0$$ and $$R^{(n)}_k= R^{(n)}_{k-1}+ \xi_k I\{R^{(n)}_{k-1}+_\xi< n\}$$, $$k\in\mathbb{N}$$. Let $$M_n$$ denote the number of jumps in the process $$\{R^{(n)}_k: k\in\{0\}\cup\mathbb{N}\}$$. The main aim of the paper is to investigate the asymptotic behaviour of $$M_n$$ as $$n\to\infty$$. Here is a sample result.
Theorem. If $$P(\xi_1\geq n)\sim L(n)/n^\alpha$$ for some function $$L$$ slowly varying at $$\infty$$ and some $$\alpha\neq ]0,1[$$, then $M_nL(n)/n^\alpha@>D>>\int^\infty_0 e^{-U_t} dt,$ where $$\{U_t: t\geq 0\}$$ is a drift-free subordinator with Lévy measure $$\nu(dt)= e^{-t/\alpha}/(1- e^{-t/\alpha})^{\alpha+1}dt$$, $$t> 0$$. The authors apply their results to derive limiting theorems for the number of collisions that take place in certain beta-coalescent processes until a single block is formed.

### MSC:

 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems 05C05 Trees 60E07 Infinitely divisible distributions; stable distributions
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### References:

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