Preliminary Exam: Probability 9:00am 2:00pm, Friday, January 6, 202 The exam lasts from 9:00am until 2:00pm, with a walking break every hour. Your goal on this exam should be to demonstrate mastery of probability theory and maturity of thought. Your arguments should be clear, careful and complete. The exam consists of six main problems, each with several steps designed to help you in the overall solution. If you cannot justify a certain step, you still may use it in a later step. On you work, label the steps this way: i, ii,... On each page you turn in, write your assigned code number instead of your name. Separate and staple each main part and return each in its designated folder.
Question. Let X, X 2,... be i.i.d. random variables with P { X > x } = e x x > 0 and let M n = max j n X j. i. 6 points Show that lim sup n X n ln n = a.s. M ii. 6 points Show that lim n n ln n = a.s. Question 2. Let {X n, n } be a sequence of i.i.d. symmetric random variables such that and P { X > x } = x p ln x P { X = 0 } = e p, for all x e where p, 2 is a constant. Prove the following statements. i. 3 points E X p =. ii. 3 points Let Y k = X k l { Xk k ln k /p }. Prove that P X k Y k i.o. = 0. iii. 5 points Verify that EY k = 0 and k=3 Y k Var /p <. k ln k [Hint: You may use the following inequality: For all n 3 and [ n lnn ] /p y [ n ln n ] /p, we have where 0 < C < is a constant.] m=n m ln m 2/p C yp 2 ln y, iv. 3 points Let S n = X + + X n for n. Prove that lim n S n n ln n /p = 0, a.s. 2
Question 3. Let {ξ n, η n, n } be a sequence of i.i.d. N0, random variables defined on a probability space Ω, F, P. i 5 points Prove that for every t R =,, the random series converges almost surely. n= sinnt cosnt ξ n + η n n n 2. ii 5 points Denote the limit in 2. by Xt. Show that Xt is a normal random variable. [Hint: You can either use characteristic functions or show that the series 2. also converges in L 2 Ω, F, P.] iii. 3 points In fact {Xt, t R} is a Gaussian process [you do not need to prove this fact]. Verify that for every s, t [0, ], that E [ Xt Xs 2] = n= 2 n 2 cosnt s. 2.2 iv. 4 points For any s, t [0, ] such that t s, let k be the integer such k + < t s k. By breaking the series in 2.2 into two sums k n= + n=k+, show that E [ Xt Xs 2] C s t for all s, t [0, ], 2.3 where 0 < C < is a constant. [Hint: You may use the inequalities that cos x x 2 and cos x 2 for all x R, as well as the fact that n=m n 2 m as m.] v. 3 points Apply ii and 2.3 to show that for every integer m, there is a constant C m such that E Xt Xs 2m C m t s m for all s, t [0, ]. Further, show that, for any γ 0, /2, for almost every ω Ω there is a constant Cω such that Xq Xr Cω q r γ for all q, r Q 2 [0, ], where Q 2 = {k2 n : k, n 0} denotes the set of dyadic rationals. [This implies that Xt has a modification which is uniformly Hölder continuous on [0, ] of any order γ < /2. But you do not need to prove this last statement.] 3
Question 4. Let {X n, n 0} be a sequence of i.i.d. µ < 0. Let S 0 = 0, S n = X + + X n, F n = σx,..., X n and W = sup S n. n 0 Nµ, σ 2 random variables with i. 4 points Prove that P { W < } =. ii. 5 points Compute E e λs n+ F n, where λ R. iii. 5 points Show that there exists a unique λ 0 > 0 such that {e λ 0S n, n 0} is a positive martingale. iv. 5 points Show that for every a >, P e λ0w > a a and thus, for every t > 0, P W > t e λ 0t. v. 4 points Show that for every 0 < λ < λ 0, Ee λw <. Question 5. Let {X n, n } be a martingale sequence w.r.t. the filtration {F n }. i. 5 points Let n k and assume that {X nk, k } is uniformly integrable. Prove that {X n, n } is also uniformly integrable. ii. 5 points Let N be a stopping time relative to the filtration {F n } and define Y k = X N k = X N l {N k} + X k l {N>k}. Prove that if E X N < and lim k E X k l {N>k} = 0, then {Yk, k } is uniformly integrable. iii. 3 points Use Part i to show that the 2nd condition in Part ii can be weakened to lim inf E X k l {N>k} = 0. You may use the fact that {Yk, k } is a martingale w.r.t. k {F k } without proof. 4
Question 6. Let {Bt, t 0} be a real-valued standard Brownian motion with respect to the canonical filtration {F t, t 0}. i. 5 points For each t > 0 we define the process { Bs : 0 s t} by Bs = Bt s Bt. Prove that { Bs : 0 s t} = {Bs : 0 s t} in distribution. [That is, the two processes have the same finite-dimensional distributions.] ii. 4 points For t > 0, denote M B t M B t = M B t Bt almost surely. = max 0 s t Bs and M B t = max Bs. Prove that 0 s t iii. 4 points Show how to conclude that Mt B Bt = Mt B in distribution. Is it true that Mt B Bt = Bt in distribution? iv. 5 points Let 0 < t < T be fixed. Define τ = inf{u t : Bu = Mt B } and ρ = inf{u t : Bu = 0}. Prove Pτ T = Pρ T by using iii and the reflection principle. 5