Course in Probability Theory

1 6 8 1 CHARACTERISTIC FUNCTION10 . Prove the following form of the inversion formula (due to Gil-Palaez) :1 1 T eitx f(-t) - e-itx f (t)2 {F(x+) + F(x-)} = 2 + Jim dt .Trx a 27rit[Hmrr : Use the method of proof of Theorem 6 .2 .1 rather than the result .]11. Theorem 6 .2 .3 has an analogue in L 2 . If the ch .f. f of F belongsto L 2 , then F is absolutely continuous . [Hmrr : By Plancherel's theorem, thereexists ~0 E L 2 such thatx 1 0o e -itx - 1cp(u) du = ~ f (t) dt .0 27r Jo -itNow use the inversion formula to show thatxF(x) - F(O) _ cp(u) du .]2n o* 12. Prove Theorem 6 .2 .2 by the Stone-Weierstrass theorem . [= : Cf.Theorem 6 .6 .2 below, but beware of the differences . Approximate uniformlyg l and 92 in the proof of Theorem 4 .4 .3 by a periodic function with "arbitrarilylarge" period .]13. The uniqueness theorem holds as well for signed measures [or functionsof bounded variations] . Precisely, if each i.ci, i = 1, 2, is the differenceof two finite measures such thatVt :I eitxA, (dx) = fe itx µ2(dx),then µl = /J, 2-14. There is a deeper supplement to the inversion formula (4) orExercise 10 above, due to B . Rosen . Under the condition(1 + log Ix 1) dF(x) < oo,00the improper Riemann integral in Exercise 10 may be replaced by a Lebesgueintegral . [HINT : It is a matter of proving the existence of the latter . Sinced00 fNJ~ dF(y) sin(x - y)t00t < dF(y){1 + log(1 +NIx - yl)} < oc,Jt fwe havef~Nsin(x t y)t dt =dF(y) N dt~00 Jo Jo j 000000sin(x - y)tdF(y) .