Course in Probability Theory

5.2 WEAK LAW OF LARGENUMBERS 1 1 1 3DEFWUION . Two sequences of r .v .'s {X n } and {Y n } are said to be equivalentiff( 1 ) 1: °J'{Xn 0 Y n } < 00 .nIn practice, an equivalent sequence is obtained by "truncating" in variousways, as we shall see presently .Theorem 5 .2 .1 . If {Xn } and [Y,) are equivalent, thenE(X n - Yn ) converges a .e .nFurthermore if a nf oo, then(2)PROOF.j=1Y j ) -)~ 0 a .e .By the Borel-Cantelli lemma, (1) implies that"P7 {Xn 0 Y n i.0 .} = 0 .This means that there exists a null set N with the following property :co E c2\N, then there exists no(co) such thatifn > no(w) , Xn (w) = Yn (CA) .Thus for such an co, the two numerical sequences {X, (c))} and {Y n (c ))) differonly in a finite number of terms (how many depending on co) . In other words,the seriesE(Xn (a)) - Yn (a)))nconsists of zeros from a certain point on . Both assertions of the theorem aretrivial consequences of this fact .Corollary .With probability one, the expression~` 1 nL~Xn or - ann j=1converges, diverges to +oo or -oc, or fluctuates in the same way asE Yn17or