Pitman's measure of closeness for symmetric stable distributions

This paper considers symmetric stable distributions with different exponents [gamma] (0 < [gamma] [less-than-or-equals, slant] 2), and studies Pitman's measure of closeness of sample averages (both weighted and unweighted) based on different sample sizes. The behavior of measures of concentration of such averages around the point of symmetry is also studied. It is found that while Pitman's closeness criterion is compatible with the measure of concentration for 1 [less-than-or-equals, slant] [gamma] [less-than-or-equals, slant] 2, this need not always be so for 0 < [gamma] < 1. The relationship of our results with the ones given in Blyth and Pathak (1985) is also pointed out.