P (A G) dp G P (A G)

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1 First homework assignment. Due at 12:15 on 22 September Homework 1. We roll two dices. X is the result of one of them and Z the sum of the results. Find E [X Z. Homework 2. Let X be a r.v.. Assume Y another r.v. for which P (Y = 0 or Y = 1) = 1. Prove Y σ(x) iff there exists a ϕ : R R Borel Measurable function such Y = ϕ(x). Homework 3. Let X and Y be random variables on the same probability space. Prove X and Y are independent iff for every ϕ : R R bounded measurable functions we have E [ϕ(y ) X = E [ϕ(y ). Homework 4. Assume X, Y are jointly continuous r.v. with joint distribution function f(x). Prove yf(x, y)dy R E [Y X = E [Y σ(x) = f(x, y)dy. Homework 5. Let Y σ(g). Prove Homework 6. Let X, Y L 1 (Ω, F, P) satisfying Show P(X = Y ) = 1. E [X G Y A G E [X 1 A E [Y 1 A. E [X Y = Y and E [Y X = X Homework 7. Prove the general version of Bayes s formula: Given the probability space (Ω, F, P) and let G be a sub-σ-algebra of F. Let G G. Show P (A G) dp G P (G A) = dp (1) P (A G) Homework 8. Prove the conditional variance formula where Var(X Y ) = E [X 2 Y (E [X Y ) 2. Ω R Var(X) = E [Var(X Y ) + Var (E [X Y), (2) Homework 9. Let X 1, X 2,... iid r.v. and N is a non-negative integer valued r.v. is independent of X i, i 1. Prove ( N ) Var X i = E [N Var(X) + (E [X) 2 Var(N). (3) i=1 Second homework assignment. Due at 12:15 on 29 September Homework 10. Let X t be a Poisson(1). That is a Poisson process with rate λ = 1. (See Durrett s book p. 139 if you forgot the definition.) Find: E [X 1 X 2 and E [X 2 X 1. Homework 11. Construct a martingale which is NOT a Markov chain. } Homework 12. For every i = 1,..., m let { M n (i) be a sequence of martingales w.r.t. {X n} n=1 n=1. Show M n := max M n (i) 1 i n is a submartingal w.r.t. {X n }. 1

2 Homework 13. Let ξ 1, ξ 2,... standard normal variables. (Recall in this case the moment generating function M(θ) = E [ e θξ i = e θ 2 /2.) Let a, b R and Prove (a) X n 0 a.s. iff b > 0 (b) X n 0 in L r iff r < 2b a 2. n S n := ξ k and X n := e asn bn Homework 14. Let S n := X X n, where X 1, X 2,... are iid with X 1 Exp(1). Verify M n := is a martingale w.r.t. the natural filtration F n. n! esn (1 + S n ) n+1 Third homework assignment. Due at 12:15 on 6 October Homework 15. Prove the following two definitions of λ-system L are equivalent: Definition 1. (a) Ω L. (b) If A, B L and A B then B \ A L (c) If A n L and A n A ( is A n A n+1 and A = n=1a n ) then A L. Definition 2. (i) Ω L. (ii) If A L then A c L. (iii) If A i A j =, A i L then i=1a i L. Homework 16. There are n white and n black balls in an urn. We pull out all of them one-by-one without replacement. Whenever we pull: a black ball we have to pay 1$, a white ball we receive 1$. Let X 0 := 0 and X i be the amount we gained or lost after the i-th ball was pulled. We define Y i := X i 2n i, for 1 i 2n 1, and Z i := (a) Prove Y = (Y i ) and Z = (Z i ) are martingales. (b) Find Var(X i ) =? X 2 i (2n i) (2n i)(2n i 1) for 1 i 2n 2. Homework 17. Let X, Y be two independent Exp(λ) r.v. and Z := X +Y. Show for any non-negative measurable h we have E [h(x) Z = 1 Z h(t)dt. Z 0 Homework 18. Let X 1, X 2,... be iid r.v. with P (X n = n 2 ) = 1 n 2 and P ( X n = n2 S n := X X n. Show (a) lim n S n /n =. 2 n 2 1 ) = 1 1 n 2. Let

3 (b) {S n } is a martingale which converges to a.s.. Homework 19. (This was withdrawn on this week and it is reassigned next week.) A player s winnings per unit stake on game n are ξ n, where {ξ} n=1 are i.i.d. r.v. P (ξ n = 1) = p and P (ξ n = 1) = q := 1 p, where surprisingly, p (1/2, 1), is p > q. That is with probability q < 1/2 the player losses her stake and with probability p she gets back twice of her stake. Let C n be the player s stake on game n. We assume C n is previsible, is C n+1 F n := σ (ξ 1,..., ξ n ) for all n. Further, we assume 0 C n Y n 1, where Y n 1 is the fortune of the player at time n. We call α := p log p + q log q + log 2 the entropy. Prove (a) log Z n nα is a supermartingale. This means the rate of winnings E [log Y n log Y 0 nα (b) There exists a strategy for which log Z n nα is a martingale. Fourth homework assignment. Due at 12:15 on 13 October Homework 20. (Reassigned from last week) A player s winnings per unit stake on game n are ξ n, where {ξ} n=1 are i.i.d. r.v. P (ξ n = 1) = p and P (ξ n = 1) = q := 1 p, where surprisingly, p (1/2, 1), is p > q. That is with probability q < 1/2 the player losses her stake and with probability p she gets back twice of her stake. Let C n be the player s stake on game n. We assume C n is previsible, is C n+1 F n := σ (ξ 1,..., ξ n ) for all n. Further, we assume 0 C n Y n 1, where Y n 1 is the fortune of the player at time n. We call α := p log p + q log q + log 2 the entropy. Prove (a) log Y n nα is a supermartingale. This means the rate of winnings E [log Y n log Y 0 nα (b) There exists a strategy for which log Y n nα is a martingale. Homework 21. Let X, Z be R d -valued r.v. defined on the (Ω, F, P). Assume E [ e it X+is Z = E [ e itx E [ e isz, s, t R n. Prove X, Z are independent. Homework 22. Let X N (µ, Σ) in R n. Let X 1 := (X 1,..., X p ) and X 2 := (X p+1,..., X n ). Let Σ, Σ 1 and Σ 2 the covariance matrix of X, X 1 and X 2 respectively. Prove X 1 and X 2 are independent if and only if ( ) Σ1 0 Σ =. 0 Σ 2 Homework 23. Let Y N (µ, Σ) and let B be a non-singular matrix. Find the distribution of X = B Y. Homework 24. Let Y N (µ, Σ) in R 2, Y = (Y 1, Y 2 ) and let a 1, a 2 R. Find the distribution of a 1 Y 1 + a 2 Y 2. Fifth homework assignment. Due at 12:15 on 20 October Homework 25. Construct a random vector (X, Y ) such both X and Y are one-dimensional normal distributions but (X, Y ) is NOT a bivariate normal distribution. Homework 26. Let X = (X 1,..., X d ) be standard multivariate normal distribution. Let Σ be an n n positive semi-definite, symmetric matrix. and let µ R d. Prove there exists an affine transformation T : R d R d, such T (X) N (µ, Σ). 3

4 Homework 27. Let X be the height of the father and Y be the height of the son in sample of father-son pairs. Then (X, Y ) is bivariate normal. Assume E [X = 68 (in inches), E [Y = 69, σ X = σ Y = 2, ρ = 0.5, where ρ is the correlation of (X, Y ). Find the conditional distribution of Y given X = 80 (6 feet 8 inches). (That is find the parameters in Y N (µ Y, σ 2 (Y )).) Homework 28 (Extension of part (iii) of Doob s optional stopping Theorem). Let X be a supermartingale. Let T be a stopping time with E [T <.(Like in part (iii) of Doob s optional stopping Theorem.) Assume there is a C such Prove E [X T E [X 0. E [ X k X k 1 F k 1 (ω) C, k > 0 and for a.e. ω. Homework 29. Let X n be a discrete time birth-death process with probabilities with transition probabilities p(i, i + 1) = p i, p(i, i 1) = 1 p i =: q i, p 0 = 1 We define the following function:, g : N R +, g(k) := 1 + k 1 j q i j=1 i=1 p i (a) Prove Z n := g(x n ) is a martingale for the natural filtration. (b) Let 0 < i < n be fixed. Find the probability a process started from i gets to n earlier than to 0. Sixth homework assignment. Due at 12:15 on 3 November Homework 30. Let {ε n } be an iid sequence of real numbers satisfying P (ε n = ±1) = 1 2. Show n ε na n converges alsmost surely iff a 2 n <. Homework 31. Let X = (X n ) be an L 2 random walk is a martingale. Let σ 2 be the variance of the k-th increment Z k := X k X k 1 for all k. Prove the quadratic variance is A n = nσ 2. Homework 32. Prove the assertion of Remark 5.6 from File C. Homework 33. Let M = (M n ) be a martingale with M 0 = 0 and M k M k 1 < C for a C R. Let T 0 be a stopping time and we assume E [T. Let U n := n (M k M k 1 ) 2 1 T k, V n := 2 (M i M i 1 ) (M j M j 1 ) 1 T j. 1 i<j n U := (M k M k 1 ) 2 1 T k, V := 2 (a) M 2 T n = U n + V n and M 2 T = U + V. 1 i<j (M i M i 1 ) (M j M j 1 ) 1 T j. Prove (b) Further, if E [T 2 < then lim n U n = U a.s. and E [U < and E [V n = E [V = 0. (c) Conclude lim n E [M 2 T n = E [M 2 T. Homework 34 (Wald equalities). Let Y 1, Y 2,... be iid r.v. with Y i L 1. Let S n := Y Y n and we write µ := E [Y i. Given a stopping time T 1 satisfying: E [T <. Prove 4

5 (a) E [S T = µ E [T. (4) (b) Further, assume Y i are bounded ( C i R s.t. Y i < C i ) and E [T 2 <. We write σ 2 := Var(Y i ). Then E [ (S T µt ) 2 = σ 2 E [T. (5) Hint: Introduce an appropriate martingale and apply the result of the previous exercise. Sevens homework assignment. Due at 12:15 on 10 November Homework 35 (Branching Processes). You might want to recall what you have learned about Bransching Processes. (See File C of the course "Stochastic Processes".) A Branching Process Z = (Z n ) n=0 is defined recursively by a given family of Z + valued iid rv. { X (n) } k as follows: k,n=1 Z 0 := 1, Z n+1 := X (n+1) X (n+1) Z n, n 0. Let µ = E [ X (n) k and Fn = σ(z 0, Z 1,... Z n ). We write f(s) for the generating function. is Further, let f(s) = P ( X (n) k = l ) s p l l=0 }{{} l for any k, n. {extinction} := {Z n 0} = { n, Z n = 0} {explosion} = {Z n }. let q := P ([extinction) :=. Recall we learned q is the smaller (if there are two) fixed point of f(s). That is q is the smallest solution of f(q) = q. Prove (a) E [Z n = µ n. Hint: Use induction. (b) For every s 0 we have E [ s Z n+1 F n = f(s) Z n. Explain why it is true q Zn lim Z n = Z exists a.s. n (c) Let T := min {n : Z n = 0}. (T = if Z n > 0 always.) is a martingale and (d) Prove q = E [ q Z T = E [ q Z 1 T = + E [ q ZT 1 T <. (e) Prove E [ q Z 1 T = = 0. (f) Conclude if T (ω) = then Z =. (g) Prove P (extinction) + P (explosion) = 1. (6) Homework 36 (Branching Processes cont.). Here we assume Prove µ = E [ X (n) k < and 0 < σ 2 := Var(X (n) k ) <. (a) M n = Z n /µ n is a martingale for the natural filtration F n (b) E [ Z 2 n+1 F n = µ 2 Z 2 n + σ 2 Z n. Conclude M is bounded in L 2 µ > 1. 5

6 (c) If µ > 1 then M := lim n M n exists (in L 2 and a.s.) and Var(M ) = σ 2 µ(µ 1). Homework 37 (Branching Processes cont.). Assume q = 1 (q was defined in the one but last exercise as the probability of extinction). Prove M n = Z n /µ n is NOT a UI martingale. Eights homework assignment. Due at 12:15 on 24 November Homework 38. Let X 1, X 2,... be iid rv. with continuous distribution distribution function. Let E i be the event a record occurs at time n. That is E 1 := Ω and E n := {X n > X m, m < n}. Prove {E i } i=1 independent and P (E i) = 1 i. Homework 39 (Continuation). Let E 1, E 2,... be independent with P (E i ) = 1/i. Let Y i := 1 Ei and N n := Y Y n. (In the special case of the previous homework, N n is the number of records until time n.) Prove (a) Y k 1/k log k converges almost surely. N (b) Using Krocker s Lemma conclude lim n = 1 a.s.. n log n (c) Apply this to the situation of the previous exercise to get an estimate on the number of records until time n. Homework 40. Let C be a class of rv on (Ω, F, P). Prove the following assertions (1) and (2) are equivalent: 1. C is UI. 2. Both of the following two conditions hold: (a) C is L 1 -bounded. That is A := sup {E [ X : X C} < AND (b) ε > 0, δ > 0 s.t. F F and P (F ) < δ = E [ X ; F < ε. Homework 41. Let C and D be UI classes of rv.. Prove C + D := {X + Y : X C and Y D} is also UI. Hint: use the previous exercise. Homework 42. Let C be a UI family of rv.. Let us define D := {Y : X C, G sub-σ-algebra of F s.t. Y = E [X G}. Prove D is also UI. Ninth homework assignment. Due at 12:15 on 1 December Homework 43. Let X 1, X 2,... be iid. rv. with E [X + = and E [X <. (Recall X = X + X and X +, X 0.) Use SLLN to prove S n /n a.s., where S n := X X n. Hint: For an M > 0 let Xi M := X i M and Sn M := Xn M + + Xn M. Explain why lim S M n n /n E [ Xi M and lim inf n S n/n lim n S M n /n. Homework 44. Let X 1, X 2,... be iid rv with E [ X i <. Prove E [X 1 S n = S n /n. (This is trivial intuitively from symmetry, but prove it with formulas.) Homework 45. Let C be a class of random variables of (Ω, F, P). Then the following condition implies C is UI: If p > 1 and A R such E [ X p < A for all X C. (L p bounded for some p > 1.) 6

7 Homework 46. Let C be a class of random variables of (Ω, F, P). Then the following condition implies C is UI: Y L 1 (Ω, F, P), s.t. X C we have X(ω) Y (ω). (C is dominated by an integrable (non-negative) r.v..) Homework 47. (Infinite Monkey Theorem) Monkey typing random on a typewriter for infinitely time will type the complete works of Shakespeare eventually. Tenth homework assignment. Due at 12:15 on 8 December Homework 48 (Azuma-Hoeffding Inequality). (a) Assume Y is a r.v. which takes values from [ c, c and E [Y = 0 holds. Prove for all θ R we have E [ e ( ) 1 θy cosh(θc) exp 2 θ2 c 2. Hint: Let f(z) := exp(θz), z [ c, c. Then by the convexity of f we have f(y) c y 2c f( c) + c + y 2c f(c). (b) Let M be a martingale with M 0 = 0 such or a sequence of positive numbers {c n } n=1 we have M n M n 1 c n, n. Then the following inequality holds for all x > 0: ( ) ( P sup M k x exp 1 ) n k n 2 x2 / c 2 k. This exercise is from the Williams book. Hint: Use submartingale inequlaity as in the proof of LIL. Then present M n (in the exponent) like a telescopic sum, then use the orthogonality of martingale increments. Use part (a), then find the minimum in θ of the expression in the exponent. Homework 49 (Exercise for Markov chain CLT). Consider the following Markov chain X = {X} n=0 : The state space is Z. The transition probabilities are as follows: p(0, 1) = p(0, 1) = 1. For an arbitrary 2 x N + := N \ {0} we have (a) Find the stationary measure π for X. Definitions p(x, x + 1) = p(x, 0) = 1 2, and p( x, x 1) = p( x, 0) = 1 2. Define the operator P : L 1 (Z, π) L 1 (Z, π) by (P g)(i) := p(i, j)g(j) and let I be the identity on L 1 (Z, π). Basically P is an infinite matrix and g is an infinite column vector and the action of the operator P on g is the product of this infinite matrix P with the infinite column vector g like (P g)(i), i Z. Let f : Z R be an arbitrary function satisfying the following conditions: x Z, f(x) = f( x), and a < 2 s.t. f(x) < a x for all x large enough. For example: polynomials of the form f(x) = n b 2i 1 x 2i 1 (b) Construct an U L 2 (π) such ((I P ) U)(i) = f(i). (c) From now on we always assume f(x) = x 3. Determine i=1 j Z σ 2 := E π [ (U(X1 ) U(X 0 ) + f(x 0 )) 2. (d) Prove P ( 3σ n f(x 1 ) + + f(x n ) 3σ n) 0.99 for sufficiently large n. 7