Resumen de Limit distributions of Studentized means

G.P. Chistyakov, Friedrich Götze

• Let X,Xj,j∈N, be independent, identically distributed random variables with probability distribution F. It is shown that Student's statistic of the sample {Xj}nj=1 has a limit distribution G such that G({−1,1})≠1, if and only if: (1) X is in the domain of attraction of a stable law with some exponent 0<α≤2; (2) \EX=0 if 1<α≤2; (3) if α=1, then X is in the domain of attraction of Cauchy's law and Feller's condition holds: limn→∞n\Esin(X/an) exists and is finite, where an is the infimum of all x>0 such that nx−2(1+∫(−x,x)y2F{dy})≤1. If G({−1,1})=1, then Student's statistic of the sample {Xj}nj=1 has a limit distribution if and only if \P(|X|>x),x>0, is a slowly varying function at +∞.