Course in Probability Theory

1 34 1 LAW OF LARGE NUMBERS . RANDOM SERIESClearly cr(Y,,) -+ (`(X 1 ) as n -+ oc ; hence alsof (Y 1 ) -+ (%'(X 1),and consequentlynj=1-- . e(X I ) a .e .By Theorem 5 .2.1, the left side above may be replaced by (1/n) I:;=1 X j ,proving (8) .To prove (9), we note that ~(IX 1 1) = oc implies AIX1 I/A) = oo for eachA > 0 and hence, by Theorem 3 .2 .1,G.1'(IX 1 I > An) = +oo .nSince the r .v .'s are identically distributed, it follows thatE ~(lXn I > An) = + oo .nNow the argument in the example at the end of Sec . 5.2 may be repeatedwithout any change to establish the conclusion of (9) .Let us remark that the first part of the preceding theorem is a special caseof G . D . Birkhoff's ergodic theorem, but it was discovered a little earlier andthe proof is substantially simpler .N. Etemadi proved an unexpected generalization of Theorem 5 .4 .2 : (8)is true when the (total) independence of {Xn } is weakened to pairwiseindependence (An elementary proof of the strong law of large numbers,Z. Wahrscheinlichkeitstheorie 55 (1981), 119-122) .Here is an interesting extension of the law of large numbers when themean is infinite, due to Feller (1946) .Theorem 5 .4 .3 . Let {X n } be as in Theorem 5 .4 .2 with ct (I X 11) = oc . Let{a„ } be a sequence of positive numbers satisfying the condition an /n T . Thenwe have(11) lim is" I n an= 0 a .e ., or= oo a .e .according as(12) t'{1X n I > a n } - dF(x) < oo, or = oc .nnPROOF . WritingccIXI >ank=nJak < Ixl