Process Convergence of Self-Normalized Sums of i.i.d. Random Variables Coming from Domain of Attraction of Stable Distributions
Gopal K Basak Arunangshu Biswas
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In this paper we show that the continuous version of the self-normalized process $Y_{n,p}(t)=S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p},0 < t \leq 1;p>0$ where $S_n(t)=\sum^{[nt]}_{i=1}X_i$ and $V_{(n,p)}=\left(\sum^n_{i=1}|X_i|^p\right)^{1/p}$ and $X_i i.i.d.$ random variables belong to $DA(\alpha)$, has a non-trivial distribution $\mathrm{iff } p=\alpha=2$. The case for $2>p>\alpha$ and $p\leq\alpha < 2$ 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 $\mathrm{iff } \alpha=2$ and identified the limiting process as a standard Brownian motion in sup norm.
Gopal K Basak^{1} ^{} Arunangshu Biswas^{2} ^{}
Volume 133, 2023
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