Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions |
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Authors: | GOPAL K BASAK ARUNANGSHU BISWAS |
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Institution: | 1. Stat-Math Unit, Indian Statistical Institute, Kolkata, 700 108, India 2. Deptartment of Statistics, Presidency College, Kolkata, 700 073, India
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Abstract: | 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 $S_n(t)=\sum_{i=1}^{nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X i i.i.d. random variables belong to DA(α), has a non-trivial distribution 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 Csörg? et al. who showed Donsker’s theorem for Y n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm. |
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