Course in Probability Theory

1 CHARACTERISTIC FUNCTIONDO h2 h +(h) -< QF (h) A QG (h)1 60*15 . For a d .f. F and h > 0, defineQF(h) = sup[F(x + h) - F(x-)] ;xQF is called the Levy concentration function of F . Prove that the sup aboveis attained, and if G is also a d .f., we haveVh > 0 : QF•G.16 . If 0 < hA < 2n, then there is an absolute constant A such thatQF(h) < - f if (t)I dt,where f is the ch .f. of F . [HINT : Use Exercise 2 of Sec . 6.2 below .]17. Let F be a symmetric d .f ., with ch .f. f > 0 then(Dr (h) _f2 dF(x) = h 00 e-ht f(t) dtoo x 0is a sort of average concentration function . Prove that if G is also a d .f. withch .f. g > 0, then we have Vh > 0 :(PF*G(h) < cOF(h) A ip0(h) ;1 - cPF*G(h) < [1 - coF(h)] + [1 - cPG(h)] .*18 . Let the support of the p .m. a on ~RI be denoted by supp p . Provethatsupp (j.c * v) = closure of supp . + supp v ;supp (µ i * µ2 * . . . ) = closure of (supp A l + supp / 2 + • )where "+" denotes vector sum .6.2 Uniqueness and inversionTo study the deeper properties of Fourier-Stieltjes transforms, we shall needcertain "Dirichlet integrals" . We begin with three basic formulas, where "sgna" denotes 1, 0 or -1, according as a > 0, = 0, or < 0 .ry sin ax(1) dy>0:0