On the functional limits for partial sums under stable law

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 $\alpha\in (1,2]$.