An asymptotic behavior of the mean of some exponential functionals of a random walk
The author studies the limiting behaviour of expressions of the type \[ \mathbb{E} \left(f(e^{S_n})h \left( \sum^n_{i=1} W(S_i) \right) \right) \sim cn^{-3/2} \tag{*} \] for \(n\to\infty\). Here, \(\{\xi_i\}_i\) is a sequence of i.i.d. random variables (centered, with absolute moments up to order 3, and some exponential moment), \(S_i= \xi_1+ \cdots +\xi_i\), and \(f,g: [0,\infty) \to \mathbb{R}\) and \(W:\mathbb{R} \to[0, \infty)\) are continuous functions. Moreover, \(f,g\) satisfy certain polynomial boundedness conditions and \(W\) an exponential growth condition at the origin and at infinity. As an application, the asymptotic behaviour for \(\mathbb{P} (\max_j X_j\geq n)\), \(n\) large, of a random walk \(\{X_n\}_n\) in a random environment is studied. This result simplifies and generalizes an earlier paper by \textit{V. I. Afanas'ev} [Theory Probab. Appl. 35, No. 2, 205-215 (1990); translation from Teor. Veroyatn. Primen. 35, No. 2, 209-219 (1990; Zbl 0714.60054)]. The proof of (*) is first done for the case where \(h(x)= e^{-tx}\) and \(W\equiv \infty\) on \([K, \infty)\). In order to do this, the author introduces a scale of exponentially weighted Banach function spaces of continuous functions and shows that (*) can be rewritten as a sequence of continuous linear transforms (applied to \(f)\) in this scale. In the corresponding Banach spaces, the limit of (*) exists, and the limiting function is given as a Neumann series. In a second step, the author shows that the limit is attained even pointwise. Now the general case follows through an approximation of \(h(x)\) by linear combinations of functions in \(\{e^{-nx}\}_n\) and letting \(K\to \infty\).
scientific article; zbMATH DE number 1132998