Course in Probability Theory

98 I CONVERGENCE CONCEPTSEXERCISES*1. Let g, and A be p.m .'s such that An A . Show that the conclusionin (2) need not hold if (a) f is bounded and Borel measurable and all Anand A are absolutely continuous, or (b) f is continuous except at one pointand every A„ is absolutely continuous . (To find even sharper counterexampleswould not be too easy, in view of Exercise 10 of Sec . 4 .5 .)2 . Let A, ~ A when the ,a 's are s .p.m .'s . Then for each f E C andeach finite continuity interval I we have fl f dA„ -). rt f dµ .*3 . Let it, and It be as in Exercise 1 . If the f„ 's are bounded continuousfunctions converging uniformly to f, then f f „ dl-t,, -* f f d A .*4 . Give an example to show that convergence in dist . does not imply thatin pr. However, show that convergence to the unit mass 8Q does imply that inpr. to the constant a .5. A set {µa } of p.m .'s is tight if and only if the corresponding d .f.'s{F a } converge uniformly in a as x --> -oc and as x -+ +oo .6. Let the r .v .'s {X a } have the p .m .'s {A a } . If for some real r > 0,{ I Xa I'} is bounded in a, then {A a } is tight .7. Prove the Corollary to Theorem 4 .4 .4 .8 . If the r .v.'s X and Y satisfyJ'{ IX - Y I> E} < Efor some E, then their d .f.'s F and G satisfying the inequalities :(15) Vx E Z I : F(x - E) - E < G(x) < F(x + E) + E .Derive another proof of Theorem 4 .4 .5 from this .*9. The Levy distance of two s .d .f.'s F and G is defined to be the infimumof all E > 0 satisfying the inequalities in (15) . Prove that this is indeed a metricin the space of s .d .f.'s, and that F„ converges to F in this metric if and onlyif F n --'*' Fand fxdF„~ f~dF .10 . Find two sequences of p .m .'s (,a,,) and {v„ } such thatbut for no finite (a, b) is it true thatVfECK :J fdA„- / fdv„>0 ;An (a, b) - v„ (a, b) - 0 .[HINT : Let A„ = 8,,, v„ = &,, and choose {r„}, {s„} suitably .]11 . Let {An } be a sequence of p.m .'s such that for each f E CB, thesequence f ,, f dA„ converges ; then An 4 A, where A is a p.m . [HINT: If the