Limit distributions of studentized means
Let X, X-j, j is an element of N, be independent, identically distributed random variables with probability distribution F. It is shown that Student's statistic of the sample {X-j}(n)(j=1), has a limit distribution G such that G({-1, 1}) not equal 1, if and only if: (1) X is in the domain of attraction of a stable law with some exponent 0 < alpha less than or equal to 2; (2) EX = 0 if 1 < alpha < 2; (3) if alpha = 1, then X is in the domain of attraction of Cauchy's law and Feller's condition holds: lim(n-->infinity) nEsin(X/a(n)) exists and is finite, where a(n) is the infimum of all x > 0 such that nx(-2)(l + integral((-x,x))y(2)F{dy}) less than or equal to 1. If G({-1, 1}) = 1, then Student's statistic of the sample (X-j)(n)(j=1) has a limit distribution if and only if P(\X\ > x), x > 0, is a slowly varying function at +infinity.
32
1A
28-77
28-77
INST MATHEMATICAL STATISTICS