Course in Probability Theory

4 .2 ALMOST SURE CONVERGENCE; BOREL-CANTELLI LEMMA 1 75*15. Instead of the p in Theorem 4 .1 .5 one may define other metrics asfollows . Let Pi (X, Y) be the infimum of all e > 0 such that' (IX - YI > E) < E .Let p2 (X , Y) be the infimum of UT{ IX - Y I > E} + E over all c > 0. Provethat these are metrics and that convergence in pr . is equivalent to convergenceaccording to either metric .* 16 . Convergence in pr . for arbitrary r .v.'s may be reduced to that ofbounded r .v.'s by the transformationX' = arctan X .In a space of uniformly bounded r .v .'s, convergence in pr . is equivalent tothat in the metric po(X, Y) = (-P(IX - YI) ; this reduces to the definition givenin Exercise 8 of Sec . 3.2 when X and Y are indicators .17. Unlike convergence in pr ., convergence a .e. is not expressibleby means of metric . [r-nwr : Metric convergence has the property that ifp(x,,, x) -+* 0, then there exist c > 0 and ink) such that p(x,1 , x) > E forevery k .]18 . If X„ ~ X a .s ., each X, is integrable and inf„ (r(X„) > -oo, thenX„- XinL 1 .19. Let f„ (x) = 1 + cos 27rnx, f (x) = 1 in [0, 1] . Then for each g E L 1[0, 1 ] we have1I f rig dx0 10fg dx,but f „ does not converge to f in measure . [HINT : This is just theRiemann-Lebesgue lemma in Fourier series, but we have made f „ > 0 tostress a point .]20 . Let {X, } be a sequence of independent r .v.'s with zero mean andunit variance . Prove that for any bounded r .v . Y we have lim, f`(X„Y) = 0 .[HINT : Consider (1'71[Y - ~ _ 1 ~ (XkY)Xk] 2 } to get Bessel's inequality c(Y 2 ) >E k=l c`(Xk 7 ) 2 . The stated result extends to case where the X„ 's are assumedonly to be uniformly integrable (see Sec . 4 .5) but now we must approximateY by a function of (X 1 , . . . , X,,,), cf. Theorem 8 .1 .1 .]4.2 Almost sure convergence; Borel-Cantelli lemmaAn important concept in set theory is that of the "lim sup" and "lim inf" ofa sequence of sets . These notions can be defined for subsets of an arbitraryspace Q .