Stochastic summability methods for domains of normal attraction of semistable laws (Q1781634)

Semistable laws arise as limit distributions of affine normalized sums or maxima of i.i.d.\ random variables along a subsequence \((k_n)\) with geometrical growth \(k_{n+1}/k_n\to c>1\). While in general the distributions do not converge along the entire sequence \(k_n=n\), for the special case of sampling sequences with \(k_n/c^n\to1\) it is shown that limitation can be achieved with certain stochastic summability methods. The summability methods can be interpreted as taking independent random samples \(T_n\to\infty\) in probability, whose mantissa for base \(c\), i.e.\ \(\mathcal M_c(T_n)=T_n/c^{\lfloor\log_cT_n\rfloor}\in[1,c)\), converges in distribution. Some examples of such summability methods are also given. Among these are logarithmic summation and certain modifications in which cases the limiting distribution is a logarithmic mixture of semistable laws.