Question

1. Let X, X1, X2, ... be random variables defined on the same space. Assume that Xn + X. Assume further that there is a random variable Y with E[Y] < o such that P(|Xn] <Y) = 1 for each n. Show that lim E[Xn] = E[X]. n-

Answer 1

ho sov. defined on space led let xn Px. N with Eryco such that each n . Pllxlcy) - 1 for then show that lim Elan) - E(X) → we have first for the to prove the theoren. care when xooi ARDO...SSSSSSSS Consider limior of the sequence Eland? which always escirts. then theire exists a subsequence xn o such that - Elo lim Elxno hence theire escirts a subsequence Lot of Land such that > by result it only then P from a of I Tim Elm) = Tim E(Xn') ao Similarly, chooning a subrequenes subrequence such that the exe that subsequence converger to lim we see that lim E(Xn) zo olhos

exists and is therefore lim Elxn) equal to zero. If yn fy then lxls" ais lle plixisy) = 1 xn-xl<ry ais. hence o asi n o » lum Ecxn) = E(X). haco