- Turkish Journal of Mathematics
- Vol: 38 Issue: 3
- Moment equalities for sums of random variables via integer partitions and Faà di Bruno's formula
Moment equalities for sums of random variables via integer partitions and Faà di Bruno's formula
Authors : Dietmar Ferger
Pages : 558-575
Doi:10.3906/mat-1301-6
View : 9 | Download : 5
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :We give moment equalities for sums of independent and identically distributed random variables including, in particular, centered and specifically symmetric summands. Two different types of proofs, combinatorial and analytical, lead to 2 different types of formulas. Furthermore, the combinatorial method allows us to find the optimal lower and upper constants in the Marcinkiewicz--Zygmund inequalities in the case of even moment-orders. Our results are applied to give elementary proofs of the classical central limit theorem (CLT) and of the CLT for the empirical bootstrap. Moreover, we derive moment and exponential inequalities for self-normalized sums.Keywords : Moments, integer partitions, Faà di Bruno's chain rule, Marcinkiewicz--Zygmund inequalities, bootstrap, self-normalized sums