Measuring the magnitude of sums
of independent random variables

Pawe Hitczenko¹
Department of Mathematics and Computer Science
Drexel University
Philadelphia, PA 19104
phitczen@mcs.drexel.edu
http://www.mcs.drexel.edu/~phitczen

Stephen Montgomery-Smith²
Department of Mathematics
University of Missouri-Columbia
Columbia, MO 65211
stephen@math.missouri.edu
http://faculty.missouri.edu/~stephen

Abstract:

This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the $p$th moment, and between the $p$th moment and the rearrangement invariant norms.

Keywords: sum of independent random variables, tail distributions, decreasing rearrangement, $p$th moment, rearrangement invariant space, disjoint sum, maximal function, Hoffmann-Jørgensen/Klass-Nowicki Inequality, Lévy Property.

A.M.S. Classification (1991): Primary 60G50, 60E15, 46E30; Secondary 46B09.