Quasicumulants and Limit Theorems in Case of the Cauchy Limiting Law

#### S.A. Molchanov, A.I. Petrov, N. Squartini

2007, v.13, Issue 3, 597-624

ABSTRACT

Starting with the trace of a random walk on $\Z^2$ on the $x$-axis, we obtain a representation of the characteristic function $f(t)$ of the trace. It follows that $f(t)\in\Delta$, where $\Delta$ is the set of real characteristic functions such that for some $k\geq2$ $\log f(t)=-a_1t+\sum_{m=2}^ka_mt^m+o(t^k)\quad (t\geq 0,t\to 0),$ where $a_1>0$ and $a_2,...,a_k$ are constants (called quasicumulants). We consider several more situations where characteristic functions having quasicumulants appear. Using the method of quasicumulants, we obtain a global limit theorem for $\P(S_n=x)$ (where $S_n=\sum_{j=1}^nX_j$ and $\{X_n\}$ is a sequence of i.i.d. integer-valued symmetric random variables) as $n+\abs{x}\to\infty$ as well as an asymptotic expansion for this probability; the leading term in these results is a properly normalized Cauchy density.

Keywords: trace process,quasicumulants,global limit theorem