JR Proof Since X and Y are independent we know that CX Y i x v ie the

Jr proof since x and y are independent we know that

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JR Proof. Since X and Y are independent, we know that C((X, Y)) = \i x v, i.e. the distribution of the ordered pair (X, Y) is equal to product measure. Given a Borel subset H C R, let B = {(x,y) G R 2 ; x + y H). Then using Fubini's Theorem, we have P(X + YeH)= P((X,Y)eB) = (/ixi/)(fl) = LR^*^) = In(lR 1 B{x,y)fJ-(dx))i^(dy) = JR viz G R; (x, y) G B} u{dy) = In ^( H - y) v ( d y) so C(X + Y)=/j,*v. If u. has density / and v has density g, then using Proposition 6.2.3, shift invariance of Lebesgue measure as in (1.2.5), and Fubini's theorem again, (n*v)(H) = !^(H-y) V (dy) = InilH-y 1 Kdx))u{dy) = J R {I H _ y f(x)X(dx))g(y)X(dy) = IR(IH f( x - v) K dx ))a{y) Hd y ) = /R ( IH f( x - y) 9{y) \(dx))\(dy) = I H (Iiif( x -y)9(y)Kdy))x(dx) = I H (f * 9)(x) X(dx), so /i * v has density given by / * g. I
9.5. EXERCISES. 113 9.5. Exercises. Exercise 9.5.1. For the "simple counter-example" with tt = N, P(w) = 2~ w for w e l l , and X n (ui) = 2 n <L,n, verify explicitly that the hypotheses of each of the Monotone Convergence Theorem, the Bounded Convergence Theorem, the Dominated Convergence Theorem, and the Uniform Integra- bility Convergence Theorem, are all violated. Exercise 9.5.2. Give an example of a sequence of random variables which is unbounded but still uniformly integrable. For bonus points, make the sequence also be undominated, i.e. violate the hypothesis of the Dominated Convergence Theorem. Exercise 9.5.3. Let X,Xi,X%,... be non-negative random variables, defined jointly on some probability triple (fl,J-, P), each having finite ex- pected value. Assume that lim n _ 00 X„(w) = X(ui) for all cje.fl. For n,if 6 N, let Y n ,K = mm(X n ,K). For each of the following statements, either prove it must true, or provide a counter-example to show it is some- times false. (a) limi^oo lim n ^oo E(Y n?K ) = B(X). (b) limn^oo limx^oo E(y n>K ) = E(X). Exercise 9.5.4. Suppose that lim. n ^oo X n (u>) = 0 for all w £ fl, but lim^oo E[X„] ^ 0. Prove that E(sup n |X n |) = oo. Exercise 9.5.5. Suppose sup n E(|X„| r ) < oo for some r > 1. Prove that {X n } is uniformly integrable. [Hint: If |X n (cA>)| > a > 0, then |X n (u;)| < \X n {u)\ r / a r ~\] Exercise 9.5.6. Prove that Theorem 9.1.6 implies Theorem 9.1.2. [Hint: Suppose \X n \ < Y where E(Y) < oo. Prove that {X n } satisfies (9.1.4).] Exercise 9.5.7. Prove that Theorem 9.1.2 implies Theorem 4.2.2, assum- ing that E|X| < oo. [Hint: Suppose {X n } / X where E|X| < oo. Prove that {Xn) is dominated.] Exercise 9.5.8. Let fl = {1,2}, with P({1}) = P({2}) = \, and let F t ({l}) = t 2 and F t ({2}) = t A for 0 < t < 1. (a) What does Proposition 9.2.1 conclude in this case? (b) In light of the above, what rule from calculus is implied by Proposi- tion 9.2.1? Example 9.5.9. Let X 1 ,X 2 ,--. be i.i.d., each with P(Xi = 1) = P(X t = -1) = 1/2.
114 9. MORE PROBABILITY THEOREMS. (a) Compute the moment generating functions Mxi^s). (b) Use Theorem 9.3.4 to obtain an exponentially-decreasing upper bound on P ( i ( X i + . . . + * „ ) > 0.1). Exercise 9.5.10. Let Xi,X2,.-. be i.i.d., each having the standard normal distribution N{0, 1). Use Theorem 9.3.4 to obtain an exponentially- decreasing upper bound on P (^(-Xi + • • • + X n ) > O.l). [Hint: Don't for- get (9.3.2).] Example 9.5.11. Let X have the distribution Exponential(5), with density fx(x) = 5 e~ bx for x > 0 (with fx{x) = 0 for x < 0).

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