The localization of random discrete measures (Q1595699)

Consider a double indexed sequence \(Y^{(n)}_i\), \(1\leq i\leq n\), \(n\geq 1\), of nonnegative random variables satisfying condition (1): their sum over \(i\) is one for each \(n\geq 1\). The author studies conditions under which (1) is `preserved' in the following limiting procedure. Rearrange, for fixed \(n\), the variables \(Y^{(n)}_i\) in a nonincreasing order \(Y_{1,n}\geq Y_{2,n}\geq\cdots\geq Y_{n,n}\) and set \(Z_{k,n}= Y_{1,n}+ Y_{2,n}+\cdots+ Y_{k,n}\). Conditions are sought under which the limit as \(k\to+\infty\) of the limsup as \(n\to+\infty\) is one of the quantities \(E(Z_{k,n})\), \(P(Z_{k,n}\geq 1-a)\) with \(a>0\) arbitrarily small, and \(Z_{k,n}\) equals one, the last one with probability one. Particular attention is paid to the case when \(Y^{(n)}_i= X_i/S_n\), where \(S_n= X_1+ X_2+\cdots+ X_n\) and the positive reandom variables \(X_i\), \(1\leq i\leq n\), are independent and identically distributed. In the special case when the \(X_i\) are regularly varying, a nearly complete solution is obtained.