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On the distribution of random walk local time. (English) Zbl 0608.60065
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 $$| \xi_ 1|^{2+m+\delta}<\infty$$.
Reviewer: B.Kryžienė

##### MSC:
 60G50 Sums of independent random variables; random walks 60J55 Local time and additive functionals 60F10 Large deviations
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