Course in Probability Theory

5 .4 STRONG LAW OF LARGENUMBERS 1 1297. For arbitrary {X,, }, ifE (-f' '(IX,, I) < 00,nthen E„ X n converges absolutely a .e .8. Let {Xn }, where n = 0, ±1, ±2, . . ., be independent and identicallydistributed according to the normal distribution c (see Exercise 5 of Sec . 4 .5) .Then the series of complex-valued r .v .'s00 einxX nn=1where i = and x is real, converges a .e. and uniformly in x . (This isWiener's representation of the Brownian motion process .)*9 . Let {X n } be independent and identically distributed, taking the values0 and 2 with probability 2 each ; then00 X nn=1converges a .e. Prove that the limit has the Cantor d .f. discussed in Sec . 1 .3 .Do Exercise 11 in that section again ; it is easier now .*10 . If E n ±X n converges a .e. for all choices of ±1, where the Xn 's arearbitrary r .v .'s, then >n Xn 2 converges a.e . [HINT : Consider >n rn (t)X, (w)where the rn 's are coin-tossing r .v .'s and apply Fubini's theorem to the spaceof (t, cw) .]3n5 .4 Strong law of large numbersTo return to the strong law of large numbers, the link is furnished by thefollowing lemma on "summability" .Kronecker's lemma . Let {xk} be a sequence of real numbers, {ak} asequence of numbers > 0 and T oo . ThenPROOF .xn- < converges -n anJ=For 1 < n < oc letnnxj --* 0 .