On the functional limits for partial sums under stable law

Details On the functional limits for partial sums under stable law On the functional limits for partial sums under stable law

2010-06-05

arxiv.org/abs/1006.1073v1

Statistics and Probability Letters 79 (2009) 1818--1822

Mathematics

Probability

Scientific paper

10.1016/j.spl.2009.05.004

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 $$(1/{\log N})\sum_{n\le N}(1/n)\mathbf {I}\bigl\{s_n(t)\le x\bigr\} \to G_t(x)\quad \text{a.s.}$$ if and only if $(1/{\log N})\sum_{n\le N} (1/n)\pp\bigl(s_n(t)\le x\bigr) \to G_t(x)$, 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 $\alpha\in (1,2]$.

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