Limiting behaviour of sums and the term of maximum modulus

Suppose that {Xn: n ^ 1} are independent and identically distributed random variables with common continuous distribution function F. Set Sn = Xt +... + Xn and Mn = V" * i, and let X[l) be the term of maximum modulus, i.e. the Xt among Xu...,Xn for which | Xt | is largest. The influence of the extreme terms on the sample sum is studied by examining the behaviour of Sn/X[l) and Sn/Mn. The main results centre about conditions for these quantities to converge to 1 in probability and almost surely. Related results deal with ratios of order statistics and ratios of record values of {Xn}. A novel feature of our approach is to study the behaviour of {SB} between successive record values of {\Xn\}. 1. Introduction an