Course in Probability Theory

3 .2 PROPERTIES OF MATHEMATICAL EXPECTATION 1 45(xii)Fatou's lemma . If X, > 0 a .e . on A, thenA ( f lim X,) &l' < lim X, 1 d:-Z'.n-+oc n-+00 ALet us prove the following useful theorem as an instructive example .Theorem 3 .2.1 .We have00 00(6) E (IXI >- n) < "( IXI) < 1 + '( IXI >_ n)I1=1 n=1so that (`(IXI) < oo if and only if the series above converges .PROOF . By the additivity property (iii), if A 11 = { n < IXI < n + 1),dGJ' .n=0 nnHence by the mean value theorem (vi) applied to each set A n0000(`°(IXI) = Y,'(7 ) ng'(An) < ( IXI) (n + 1)~P(An) = 1 + n~P(An)jn=0n=0n=0It remains to show00(8) n_,~P(An) _ ( IXI > n),n=0n=0finite or infinite . Now the partial sums of the series on the left may be rearranged(Abel's method of partial summation!) to yield, for N > 1,N(9) En{J~'(IXI > n) -'(IXI > n + 1)}n=0f IXI0000n=1- - 1)}"P(IXI >- n) -N~(IXI > N+ 1)Thus we haveNn=1NN'(IXI > n) - NJ'(IXI > N + 1) .(10) Ln'~,)(An) < ° ( IXI > n) .~/(An)+N,I'(IXI > N+ 1) .n=1n=1n=1