Limit theorems in the boundary hitting problem for a multidimensional random walk (Q2773619)

Let \(\{\xi(i)\}_{i=1}^\infty\) be i.i.d.\ random vectors in \(\mathbb R^d\) with non-degenerate distribution. Let \(S(n)\) be a random walk \(S(n)=\xi(1)+\cdots+\xi(n)\). For a fixed set \(V\subseteq\mathbb R^d\) whose closure does not contain the origin, the authors consider the distributions connected with the time and place of the first entrance of the trajectory \(\{S(n)\}_{i=1}^\infty\) into the set \(tV\) as \(t\to\infty\). Put NEWLINE\[NEWLINE \eta=\eta(tV)=\min\{n\geq 1:S(n)\in tV\}\quad \text{and}\quad \chi=\chi(tV)=\xi(\eta)(1-p), NEWLINE\]NEWLINE where \(p=\inf\{u\in(0,1]:S(\eta-1)+p\xi(\eta)\in tV\}\). Asymptotic representations in the domain of normal deviations as well as in the domain of large deviations for the joint distribution of \((\eta(tV),S(\eta(tV))-\chi(tV),\chi(tV))\) are suggested. The authors give local and integral versions of the corresponding results.