Process Convergence of Self-Normalized Sums of i.i.d. Random Variables Coming from Domain of Attraction of Stable Distributions

Proceedings Mathematical Sciences

Gopal K Basak²  Arunangshu Biswas¹ 

[1] Department of Statistics, Presidency University, / College Street, Kolkata 00 0, India$$;Stat-Math Unit, Indian Statistical Institute, 0 B T Road, Kolkata 00 0, India$$

关键词: Domain of attraction;    process convergence;    self-normalized sums;    stable distributions.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper we show that the continuous version of the self-normalized process $Y_{n,p}(t)=S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p},0 < t ≤ 1;p>0$ where $S_n(t)=sum^{[nt]}_{i=1}X_i$ and $V_{(n,p)}=left(sum^n_{i=1}|X_i|^pight)^{1/p}$ and $X_i i.i.d.$ random variables belong to $DA(𝛼)$, has a non-trivial distribution $mathrm{iff}$ 𝑝=𝛼=2. The case for 2>𝑝>𝛼 and 𝑝 ≤𝛼 < 2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörgő et al. who showed Donsker’s theorem for $Y_{n,2}(cdot p)$, i.e., for $p=2$, holds $mathrm{iff}$ 𝛼=2 and identified the limiting process as a standard Brownian motion in sup norm.

【 授权许可】

Unknown