Asymptotic Behaviors for Random Geometric Series

In this paper, we consider asymptotic behaviors for random geometric series. We first study the convergence rates in the central limit theorem, i.e., the Berry–Esseen bound and Edgeworth expansions, and precise deviations. Then we define a bounded linear operator from the path space of random walk to the path space of the random geometric series and establish the functional central limit theorem, the functional law of iterated logarithm, and functional large deviation principles for the random geometric series.