2. Let ψ(t) be a c.f. of some r.v. X. For a < 0, if ψa(t) is also a c.f., then X must be degenerate.
3. Find two dependent r.v.s X, Y such that ψX+Y (t) = ψX (t)ψY (t).
(Hint: choose X = Y and use c.f. to find one example.)
Remark: It is known that if X, Y are i.i.d. with the same d.f. F , then X + Y has distribution F ∗ F (i.e. convolution of F with F ). This exercise illustrates that the same conclusion may still hold when X, Y have the same d.f. F, but not independent.
4. (a) Find the c.f. ψX(t) if P(X = ±α) = 1/2.
(b) Show by iteration of an elementary trigomonetric identity that
sint sint/2n n t ∞ t = cos −→ cos .
tt/2n 2j 2j j=1 j=1
(c) Utilize (b) to state a result on convergence in law of sums of independent r.v.s.
7. Let X1,X2,... be independent and let Sn = X1 + ... + Xn. Let ψj be the c.f. of Xj, and suppose ∞
that Sn → S∞ a.s. Then S∞ has c.f. ψ∞(t) = j=1 ψj(t).
9. (Fn(x) → F(x) does not imply Fn′ (x) → F′(x)) Let Xn hav d.f.
, x ∈ [0, 1]. (a) Show that Fn is indeed a d.f. and find its p.d.f. fn.
Remark. The example shows that convergence of d.f.s does not imply that converge of their p.d.f.s., which is referred to as local limit theory. Extra conditions will be required for the latter to hold.
̄ −1 n
10. Let X1, X2, ... be i.i.d. Cauchy r.v.s. Show that X = n j=1 Xj is still a Cauchy r.v.
11. (Thelimitofc.f.smaynotbeac.f.)Verifythatthec.f.correspondingtoauniformd.f.on(−n,n)is ψn(t) = sin nt/(nt), and that limn ψn(t) exists but for all t, but ψ(t) is not a c.f. Consequently, Fn ̸=⇒ a d.f.
Fn(x) = x −
(b) Show that Fn =⇒ U[0,1], but fn’s do not converge.
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