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Xk e U + , f{xx + •■■ +xk) < 2"(f(xx) + - +f(xk)) +f(k2). 15) Proof. Because the function / is convex, /(]£(*. +••■+**)) ^ ( / ( * i ) + -+/(**)); therefore / ( * , + - +**) < l{f{kxx) + - + f(kxk)). On the other hand, if 0 < x < y, then f{xy) <2"xf(y); therefore f(kx) <2"k(f(x)+f(k)). 4 47 IDENTICALLY DISTRIBUTED CASE Consequently, / ( * , + - +xk) < 2"(f(Xl) + - +f(xk) = 2»(f(xl) + */(*)) -+f(xk))+f(k2), + which was to be proved. Lemma 2. For all n > 1, (3) ' " / 7 ^ — i W , - * * ) ^ . ) ¿ti - dxn = —(\og+ x) -* rt " • 1 {x>Xi-x„)dx1 1 ■-dxn-—x(\ogx)n x (*eR+); (*-*»), i/ifl/ ¿s, i/ie rario o/ i/ie left-hand side to the right-hand side converges to 1 when X - > oo.

It follows that the series E, Y,/{t) converges. On the other hand, E Í E w J = E V , * Yt) = Epfl^i > } = E P { W > <0P}, / Í í t and the right-hand side is less than or equal to supEO^I'jE T<0' < oo- therefore P{«:card{i e D: * , ( „ ) # y,(o,)} < oo} = P { E W y , } < oo} = 1. 3 45 THE STRONG LAW OF LARGE NUMBERS Consequently, the series L,(X,/(t)) converges. , d), we still have to check that sup, E s < ( Xs/(s)\ < oo, or equivalently, that sup, |E, s f (Y, - E(Ys))/(s)\ < oo. 14), we see that 2 / _ Yt - E(YS) \ 1 < -|sup E = -|sup ' DE £ Y, ~ E(V,) (s) y, - E ( y j JS/ 4EE i / y , - E ( y , ) <Í> *7?

Uniformly integrable if the family (X,) is uniformly integrable. We will say that (J7¡, t e J) is the natural filtration of a process X indexed by J if for all t e J, &¡ is the sigma field generated by the family (Xs, s e J, i < /). Recall that according to our conventions, &¡ contains all null sets of ÍF. Clearly, every process is adapted to its natural filtration, which means that it satisfies condition a above with Sf, instead of &t. If A" is a process indexed by J and T is a stopping point with values in J, then XT will denote the random variable co >-» XT(

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