On the functional limits for partial sums under stable law

TL;DR

This paper establishes almost sure limit theorems for partial sums of independent variables under stable laws, characterizing the convergence of empirical distributions to a limiting distribution and extending to products of sums for variables attracted to stable laws.

Contribution

It provides a new almost sure limit theorem for the empirical distribution of normalized partial sums under stable law attraction, and extends results to products of partial sums for i.i.d. variables.

Findings

Almost sure convergence of empirical distribution functions to a limit distribution.

Characterization of the limit process via a logarithmic average.

Functional limit theorem for products of partial sums.

Abstract

For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\sum_{k \le [nt]} ({S_k}/{k}-\mu)$ and prove that $$(1/{\log N})\sum_{n\le N}(1/n)\mathbf {I}\left\{s_n(t)\le x\right\} \to G_t(x)\quad \text{a.s.}$$ if and only if $(1/{\log N})\sum_{n\le N} (1/n)\mathbb{P}\left(s_n(t)\le x\right) \to G_t(x)$, for some sequence $(d_n)$ and distribution $G_t$. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an $\alpha$-stable law with $\alpha\in (1,2]$.