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.