On the asymptotic normality of self-normalized sums

Abstract

Let X1, …, Xn be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let Sn(rn) denote their sum when the rn largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of Sn(rn), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when rn = r these conditions are the same, but when rn → ∞the former holds more generally.

Random Deletion Does Not Affect Asymptotic Normality or Quadratic Negligibility

“…It would be interesting to know whether (2.17) or (2.18) implies (2.2) and hence (2.1). In one dimension, and when there is no deletion, this is true and shown by Griffin and Mason [8] and Gine , Go tze, and Mason [6].…”

On bounds for moderate deviations for Student's statistic

“…Ry will not in general be finite, however it will be finite for the normal and other strongly unimodal distributions (see, for example, [10], [8], [6], [5], [11], [3]). …”

“…Self-normalized sums satisfy the CLT provided that X belongs to the domain of attraction of a normal law ([17], [5]). P. Griffin and D. Mason [10] proved that the condition is necessary in the symmetric case. E. Gine, F. Gotze and D. Mason [9] proved that the condition is necessary for nonsymmetric X as well.…”

On the central limit theorem along subsequences of noncorrelated observations

“…In this case, the question on limit distributions of T n was raised in 1973 by B. Logan, C. Mallows, S. Rice and L. Shepp [12] solving the problem for stable laws F of X n 's. After contributions [13], [9], [8] and others, a complete answer in terms of F has recently been given by G. Chistyakov and F. Gotze [6]. In particular, the central limit theorem for T n holds under weaker assumptions than that for S n (that is, a little less than finiteness of the second moment of F is required).…”