MATH 587Assignment 4Due November 30, 2004(1) LetX∈L2(Ω,F,P) and letGbe aσ-subalgebra ofF. Show that among allG-measurable r.v.’sY∈L2(Ω,F), there is a unique (a.s.) r.v.Y0which minimizeskX−Yk2.(2) (a) Suppose thatY1,Y2,...are non-negative random variables and that∑∞n=1EYn<∞. Show thatasn→∞,Yn→0 a.s.(b) Use (a) to show that ifX1,X2is a sequence of iid random variables with meanµ, varianceσ2, and a Fnite fourth moment, thenSn/n→µa.s. asn. (Hint: putYn=(Sn−nµ)4/n4,whereSn=∑ni=1Xi.) This is due to Cantelli.
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This note was uploaded on 01/25/2011 for the course STAT 235a at Stanford.