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 $$(1/\log N)\sum _{n\le N}(1/n)\mathbf{I}\left\{s_n(t)\le x\right\}\to G_t(x)\text{a.s.}$$ 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 $\alpha \in (1,2]$.

References in corpus (2)

• On the functional limits for sums of a function of partial sums
• On the functional limits for partial sums under stable law

Cited by in corpus (1)

• On the functional limits for partial sums under stable law