Course in Probability Theory

90 1 CONVERGENCE CONCEPTSinfinity such that the numbers An, (a, b) converge to a limit, say L :A g(a, b) .By Theorem 4 .3 .3, the sequence {g,,, k > 1} contains a subsequence, sayLu n t I k > 11, which converges vaguely, hence to g by hypothesis of thetheorem. Hence again by Theorem 4 .3 .1, (ii), we haveA n , (a, b) -+ A(a, b) .But the left side also -f L, which is a contradiction .EXERCISES1. Perhaps the most logical approach to vague convergence is as follows .The sequence {An, n > 1 } of s .p.m .'s is said to converge vaguely iff thereexists a dense subset D of £ such that for every a E D, b E D, a < b, thesequence (An (a, b), n > 1 } converges . The definition given before implies this,of course, but prove the converse .2 . Prove that if (1) is true, then there exists a dense set D', such thatAn (I) -+ g(I) where I may be any of the four intervals (a, b), (a, b], [a, b),[a, b] with a E D', b c D' .3. Can a sequence of absolutely continuous p .m .'s converge vaguely toa discrete p.m .? Can a sequence of discrete p .m .'s converge vaguely to anabsolutely continuous p.m.?4. If a sequence of p .m .'s converges vaguely to an atomless p.m ., thenthe convergence is uniform for all intervals, finite or infinite . (This is due toPolya .)5. Let {fn } be a sequence of functions increasing in R1 and uniformlybounded there : Supra,,, I fn WI < M < oo . Prove that there exists an increasingfunction f on 91 and a subsequence {n k } such that f nk (x) -* f (x) for everyx. (This is a form of Theorem 4 .3 .3 frequently given ; the insistence on "everyx" requires an additional argument .)6. Let {g n } be a sequence of finite measures on /31 . It is said to convergevaguely to a measure g iff (1) holds . The limit g is not necessarily a finitemeasure . But if It, (M ) is bounded in n, then g is finite .7 . If 9'n is a sequence of p .m.'s on (Q, 3T) such that ,/'n (E) convergesfor every E E ~7, then the limit is a p .m . ~' . Furthermore, if f is boundedand 9,T-measurable, thenI f d .~'n -- ' f dJ' .J s~ ~(The first assertion is the Vitali-Hahn-Saks theorem and rather deep, but itcan be proved by reducing it to a problem of summability ; see A. Renyi, [24] .