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Almost sure limit points of lightly trimmed sums when the distribution function belongs to the domain of partial attraction of a positive semi-stable law

Vasudeva R., Srilakshminarayana G.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

129--142

In this paper, we obtain the set of all almost sure limit points of lightly trimmed sums, properly normalized, when the underlying distribution function belongs to the domain of partial attraction of a semi-stable law supported on the positive half axis.

On stability of trimmed sums

Hu T.-C., Huang C.-Y., Rosalsky A.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

153--172

Let {Xn, n ≥ 1} be a sequence of i.i.d. random variables and let {an, n ≥ 1} and {bn, n ≥ 1} be sequences of constants where 0 < bn ↑ ∞. Let Xn(1), Xn(2),…, Xn(n) be a rearrangement of X1,…, Xn such that |Xn(1)| ≥ |Xn(2)| ≥ … ≥ |Xn(n)|. Consider the sequence of weighted sums Tn = Σni=1 ai Xi, n ≥ 1, and, for fixed r ≥ 1, set T(r)n = Σni=1 ai Xi I(|Xi| ≤ |X(r+1)n|), n ≥ r + 1; i.e., T(r)n is the sum Tn minus the sum of the X(k)n’s multiplied by their corresponding coefficients for k = 1,…, r. The main results provide sufficient and, separately, necessary conditions for b−1n T(r)n − kn → 0 almost surely for some sequence of centering constans {kn, n ≥1}. The current work extends that of Mori [14], [15] wherein an ≡ 1.