Estimates for overshooting an arbitrary boundary by a random walk and their applications (Q2711118)

For a sequence \(X = \{X(n)\}_{n \geq 0}\), \(X(0) = 0\), of real r.v.'s with \(E|X(n)|< \infty\), for all \(n\), and nonrandom \(x > 0\) let \(\eta(x) = \infty\) if \(\sup_k X(k) \leq x\), otherwise \(\eta(x) = \min\{ k \geq 1: X(k) > x\}\). On the event \(\{\eta(x) < \infty\}\) define the overshoot value \(\chi(x) = X(\eta(x)) - x\). Under specified conditions the upper bounds are given for \(P\{\chi(x) > t\}\) and \(E\eta(x)\) with arbitrary \(x,t \geq 0\). Note that the increments \(X(n+1) - X(n)\) need not have finite second moment, be equidistributed and/or independent. If \(X\) is a certain Markov chain (nonhomogeneous but with asymptotically homogeneous jumps, etc.), the Lévy distance between the distribution of \(\chi(x)\) and that of so called overshoot \(\chi_{\text{ac}}(\infty)\) over the infinitely distant level for some auxiliary random walk is studied. The main attention is given then to the first overshoot over certain nonconstant (nonlinear deterministic or random) boundaries \(g(x,\cdot): \mathbb{R}^{+} \to \mathbb{R}^{+}\), \(x > 0\), with \(g(x,0) = x\), namely, to analysis of weak convergence of \(\eta_g (x) = \min \{n \geq 0: X(n) > g(x,n)\}\) as \(x \to \infty\). As an application the uniform weak convergence (on some class of distributions \(F\)) is established for the overshoot \(\chi^{(F)} (x)\), above the level \(x \to \infty\), by the sums \(\xi_1^{(F)} + \cdots + \xi_n^{(F)}\) of i.i.d. r.v.'s having the common distribution \(F\). Whence related uniform renewal theorems follow in the ``homogeneous'' and Markovian cases. Corollaries for the mean time of crossing a border and two examples are provided as well.