On the distribution of random walk local time
Let \(\{\xi_{\ell}\}^{\infty}_{\ell =1}\) be i.i.d. random variables, \(E\xi_ 1=0\). \(\xi_ 1\) is assumed to have integer values, and \(| E \exp (it\xi_ 1)| =1\) if and only if t is a multiple of \(2\pi\). Let \(\nu_ k=\xi_ 1+...+\xi_ k\), \(1_ A(\cdot)\) be the indicator function of set A, and \(\phi (n,r)=\sum^{n}_{k=1}1_{\{r\}}(\nu_ k)\). The function \(\phi\) (n,r) is the local time of random walk \(\nu_ k\). It is known [the author, Teor. Veroyatn. Primen. 26, 769-783 (1981; Zbl 0474.60056)] that the process \[ {\mathfrak Z}_ n(t,x):=n^{- 1/2}\phi^{([nt],[x\sqrt{n}])} \] converges weakly as \(n\to \infty\) to Brownian local time. Limit theorems of large deviations type are proved for probabilities P(\({\mathfrak Z}_ n(1,x)=y)\) and P(\({\mathfrak Z}_ n(1,x)\geq y)\) when \(\xi_ 1\) satisfies Cramér's condition. Similar results including explicit asymptotic expansions are proved when \(\xi_ 1\) satisfies the condition E \(| \xi_ 1|^{2+m+\delta}<\infty\).
scientific article; zbMATH DE number 3984253