Abstract:

The paper presents results on the convergence $ \exp\left(i2\pi {\alpha_n \over S_n}\right) \rightarrow \exp(i2\pi U)$, as $ n \rightarrow \infty$ in distribution where $S_n$ is a random walk with zero mean and a positive finite variance. The positive real numbers $\alpha_n$ fulfill $n^{-{1\over2}}\alpha_n\rightarrow+\infty$ and $U$ is a random variable uniformly distributed on the interval $[0,1)$. The asymptotics is derived in a more general setting for a sequence of random variables $S_n$ that have either absolutely continuous distributions or distribution functions which satisfy a Berry-Esseen type condition.