Course in Probability Theory

6 .5 REPRESENTATIVE THEOREMS I 195EXERCISES1 . If f is continuous in :W' and satisfies (3) for each x E and eachT > 0, then f is positive definite .2. Show that the following functions are ch .f.'s :1 _ 1 - Itl", if Iti < 1 ;1 -+ Itl'f (t) 0, if Itl > 1 ; 0 < a < l,f(t) =1- Itl, if 0 < Itl < i4iif Iti > z .3 . If {X,} are independent r .v.'s with the same stable distribution ofexponent a, then Ek=l X k /n 'I' has the same distribution . [This is the originof the name "stable" .]4 . If F is a symmetric stable distribution of exponent a, 0 < a < 2, thenf 00 Ix J' dF(x) < oo for r < a and = oo for r > a . [HINT : Use Exercises 7and 8 of Sec . 6 .4 .]*5. Another proof of Theorem 6 .5 .3 is as follows . Show thatand define the d .f.Next show thatG on R+ by1000tdf'(t) = 1G(u) = ft d f'(t) .[0 u](1 - Itl dG(u) = f (t) .JoUHence if we setf(u,t)=(1_u) v0u(see Exercise 2 of Sec . 6 .2), thenf (t) = f[O,oc)f (u, t) dG(u) .Now apply Exercise 2 of Sec . 6 .1 .6 . Show that there is a ch .f. that has period 2m, m an integer > 1, andthat is equal to 1 - Itl in [-1, +1] . [HINT : Compute the Fourier series of sucha function to show that the coefficients are positive .]