Course in Probability Theory

74 1 CONVERGENCE CONCEPTSEXERCISES1 . X„ -~ +oc a .e . if and only if VM > 0 : 91{X, < M i .o .) = 0 .2 . If 0 < X,, X, < X E L1 and X, -* X in pr ., then X, -~ X in L 1 .3. If X n -+ X, Y„ -* Y both in pr ., then X, ± Y„ -+ X + Y, X„ Y nX Y, all in pr.*4 . Let f be a bounded uniformly continuous function in Then X,0 in pr . implies (`if (X,,)) - f (0) . [Example :-af (x)_ 1XI1+1XIas in Theorem 4 .1 .5 .]5. Convergence in LP implies that in L' for r < p .6. If X, ->. X, Y, Y, both in LP, then X, ± Y n --* X ± Y in LP . IfX, --* X in LP and Y, -~ Y in Lq, where p > 1 and 1 / p + l 1q = 1, thenX n Y n -+ XY in L 1 .7. If X n X in pr. and X n -->. Y in pr ., then X = Y a .e .8 . If X, -+ X a .e . and µn and A are the p .m .'s of X, and X, it does notfollow that /.c„ (P) - µ(P) even for all intervals P .9. Give an example in which e(X n ) -)~ 0 but there does not exist anysubsequence Ink) - oc such that Xnk -* 0 in pr.* 10 . Let f be a continuous function on M. 1 . If X, -+ X in pr., thenf (X n ) - f (X) in pr . The result is false if f is merely Borel measurable .[HINT : Truncate f at +A for large A .]11 . The extended-valued r .v . X is said to be bounded in pr. iff for eachE > 0, there exists a finite M(E) such that T{IXI < M(E)} > 1 - E . Prove thatX is bounded in pr . if and only if it is finite a .e .12 . The sequence of extended-valued r.v . {X n } is said to be bounded inpr . iff supra W n I is bounded in pr . ; {Xn ) is said to diverge to +oo in pr . iff foreach M > 0 and E > 0 there exists a finite n o (M, c) such that if n > n o , thenJT { jX„ j > M } > 1 - E . Prove that if {X„ } diverges to +00 in pr . and {Y„ } isbounded in pr ., then {X n + Y„ } diverges to +00 in pr .13 . If sup„ X n = +oc a .e ., there need exist no subsequence {X nk } thatdiverges to +oo in pr .14. It is possible that for each co, lim nXn (co) = +oo, but there doesnot exist a subsequence {n k } and a set A of positive probability such thatlimk X nk (c)) = +00 on A . [HINT : On (//, A) define X n (co) according to thenth digit of co .]