Course in Probability Theory

186 1 CHARACTERISTIC FUNCTION25 . If F 0 60, then there exists a constant A such that for every j :5{S,-1 /2= j} < An .[HINT : Use a special case of Exercise 27 below .]26 . If F is symmetric and f Ix I dF(x) < oo, thenn-13111S, = j } --+ oc .[HINT : 1 -f(t) =o(ItI) as t-+ 0 .127. If f is any nondegenerate ch . f, then there exist constants A > 0and 8 > 0 such thatf (t)I < 1 -At 2 for I t I < 8 .[HINT: Reduce to the case where the d .f. has zero mean and finite variance bytranslating and truncating .]28 . Let Q,, be the concentration function of S, = En j-1 X j , where theXD 's are independent r.v.'s having a common nondegenerate d .f . F . Then forevery h > 0,Qn (h) < An -1 / 2[HINT: Use Exercise 27 above and Exercise 16 of Sec . 6 .1 . This result is dueto Levy and Doeblin, but the proof is due to Rosen .]In Exercises 29 to 35, it or Ak is a p.m . on u41 = (0, 1] .29 . Define for each n :fu(n) = fe2mnx A(dx) .9rProve by Weierstrass's approximation theorem (by trigonometrical polynomials)that if f A , (n) = f ,,, (n) for every n > 1, then µ 1 - µ2 . The conclusionbecomes false if =?,l is replaced by [0, 1] .30 . Establish the inversion formula expressing µ in terms of the f µ (n )'s .Deduce again the uniqueness result in Exercise 29 . [HINT : Find the Fourierseries of the indicator function of an interval contained in W .]31 . Prove that I f µ (n) I = 1 if and only if . has its support in the set{eo + jn -1 , 0 < j < n - 1} for some 00 in (0, n -1 ] .*32. ,u, is equidistributed on the set {jn -1 , 0 < j < n - 1 } if and only iff A(i) = 0 or 1 according to j t n or j I n .* 33 . itk- v µ if and only if f :-, k ( ) -,, f ( ) everywhere .34 . Suppose that the space 9/ is replaced by its closure [0, 1 ] and thetwo points 0 and I are identified ; in other words, suppose 9/ is regarded as the