Proceedings Mathematical Sciences | |
Process Convergence of Self-Normalized Sums of i.i.d. Random Variables Coming from Domain of Attraction of Stable Distributions | |
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 $DA(ð›¼)$, has a non-trivial distribution $mathrm{iff}$ ð‘=ð›¼=2. The case for 2>ð‘>𛼠and ð‘ ≤𛼠< 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 CsörgÅ‘ et al. who showed Donsker’s theorem for $Y_{n,2}(cdot p)$, i.e., for $p=2$, holds $mathrm{iff}$ ð›¼=2 and identified the limiting process as a standard Brownian motion in sup norm.
【 授权许可】
Unknown
【 预 览 】
Files | Size | Format | View |
---|---|---|---|
RO201912040507029ZK.pdf | 223KB | download |