(b) Kolmogorov’s three-series theorem on convergence of sums of…

Question Answered step-by-step (b) Kolmogorov’s three-series theorem on convergence of sums of… (b) Kolmogorov’s three-series theorem on convergence of sums of IRVs. (c) Doob’s maximal Lp inequalities for martingales. (d) Method of characteristic functions in weak convergence. 2. Let (Mn) be a martingale bounded in L1, that is, supn E [|Mn|] < ∞. (a) Will (Mn) converge in distribution? (b) Will supn |Mn| < ∞ (a.s)? (c) Will (Mn) be uniformly integrable? (d) Is it possible to find a random variable M∞ such that Mn = E [M∞| Fn]? Justify your answers. 3. Let (Xn) be IRVs with zero mean such that |Xn| ≤ K for a positive constant K. Let Sn = X1 + · · · + Xn, S0 = 0, and suppose that (Sn) converges (a.s.) to a RV Y . (a) Can we assert that Sn = E [Y | X1, . . . , Xn]? (b) Is it true that E [supn S 2 n ] < ∞? Justify your answers. 4. Let (Xn) be IID Gaussian RVs with mean 0 and variance 1. Denote Sn = X1 + · · · + Xn, S0 = 0. Prove that P max k≤n Sk ≥ c ≤ e − c 2 2n , ∀c > 0. 5. Let (Xn) be IID RVs with uniform distribution on [0, 1]. We denote Mn = 2n Y k≤n Xk, n ≥ 1. Show that there is a RV Y such that Mn → Y (a.s.) and compute Y . 6. Let (Xn) be IID RVs with uniform distribution on [0, 1]. Compute the characteristic function φn = φn(θ) of Yn = E [X1| X1 + · · · + Xn] . Show that φn(θ) → φ(θ), θ ∈ R, and compute φ = φ(θ). 7. A gambler throws dice until he gets N identical outcomes in a row. The cost of one throw is q and the reward at the end is A. Find the relationship between q, A and N for the game to be fair, that is, for the expected payoff to be 0. 8. Let (Xn) be IID RVs with uniform distribution on [0, 1]. Denote Sn = X1 + · · · + Xn, S0 = 0, and τ = min{n ≥ 0 : Sn > 1}. Compute E [τ ] and E [Sτ ]Tail σ-algebra. Kolmogorov’s 0 − 1 law. (c) Give the definitions of the following convergences: (i) almost surely, (ii) in probability, (iii) in L1, (iv) weak (= in distribution). Specify all relations between these convergences. (d) Let (Xn) be a nonnegative martingale. Will it converge (i) almost surely, (ii) in probability, (iii) in L1, (iv) weakly to some random variable X∞? If needed, formulate additional (ideally, necessary and sufficient) conditions on (Xn) under which these convergences take place. (e) Method of characteristic functions in weak convergence. 2. At time 0, an urn contains 1 black ball and 1 white ball. At each time 1, 2, 3, . . . , a ball is chosen at random from the urn and is replaced together with a new ball of the same color. Just after time n, there are therefore n + 2 balls in the urn, of which Bn + 1 are black, where Bn is the number of black balls chosen by time n. Let Mn = (Bn + 1)/(n + 2), the proportion of black balls in the urn just after time n. Prove that (Mn) converges a.s. to a RV Θ and find the distribution of Θ. 3. Let (Xn) be positive IID RVs with same continuous distribution function and denote Mn = max0≤k≤n Xk. Will the series X n Xn1{Xn=Mn} converge? 4. Let (Xn ) be IRVs such that |Xn | ≤ 1 and P n Xn converges in distribution. Will this series also converge almost surely? 5. Let (Mn) be a martingale with M0 = 0 and suppose that |Mn − Mn+1| ≤ cn, for some cn ≥ 0. Prove that P(sup n Mn ≥ x) ≤ e − x 2 2a2 , x > 0, where a 2 = X n c 2 n . 6. Let (Yn) are IID Bernoulli’s RVs taking values 1 and −1 with probability 1/2. For an integer n ≥ 1 define Xn , Y0Y1 . . . Yn. What is the minimal value of P({Y0 = 1} ∩ A), where we are minimizing over the events A such that A ∈ ∩n≥1σ(Xn, Xn+1, . . .) and P(A) ≥ 1/8. 7. Let X and Y be IID RVs such that X + 1 2 Y has uniform distribution on [−1, 1]. Compute the characteristic function φ = φ(θ) of X. Will the integral I , Z R |φ(θ)|dθ be finite? 8. Let (Xn) be a symmetric random walk on integers starting at 0. Among all stopping times τ find the one that maximizes E[|Xτ |e −λτ ], where λ > 0. The answer may not be too explicit. Do the best you can Give the definitions of the following convergences: (i) almost surely, (ii) in probability, (iii) in L1, (iv) weak (= in distribution). Specify all relations between these convergences. (b) Doob’s maximal Lp inequalities for martingales. (c) Let (Xn) be a non-negative supermartingale. Will it converge (i) almost surely, (ii) in probability, (iii) in L1, (iv) weakly to some random variable X∞? If needed, formulate additional (ideally, necessary and sufficient) conditions on (Xn) under which these convergences take place. (d) Method of characteristic functions in weak convergence. 2. Let (Xn) be IID RVs such that E [X1] = 0, 0 < E X 2 1 < ∞. P For each of the items (i)-(iv) describe all sequences of real numbers (an) such that n anXn converges (i) almost surely, (ii) in probability, (iii) in L1, (iv) in L2. 3. Let (Xn) be IID RVs in L1 such that X1 ≥ 0, E [X1] = 1 and X1 6= 1. Set Yn , Yn k=1 Xk. Can we find a filtration (Fn) and a random variable Y∞ such that Yn = E[Y∞|Fn]?.Let (Xn) be IID random variables taking values in [−1, 1] and having the common mean µ = E [Xn] = 0 and the variance σ 2 = E [X2 n ] > 0. Let (an) be a sequence in (−1, 1). Define Yn , Yn k=1 (1 + akXk), n ≥ 1. Will (Yn) converge (i) almost surely, (ii) in probability(4 points) While not strictly true, people tend to view hash tables as offering constant time look up in practice. (The possibility of a bad hash function leading to many collisions is why it is not strictly constant time.) Distributed Hash Tables (DHTs), on the other hand, only offer O(log n)-time lookup where n is the number of nodes in the DHT. This is odd, since the two appear to be equivalent, i.e. simply map each entry in a traditional hash table onto a node in the DHT. What property or requirement of a DHT makes this approach impractical in practice., (iii) in L1, (iv) weakly to some random variable Y∞? Will Yn = E [Y∞| σ(X1, . . . , Xn)]? If needed, formulate additional (ideally, necessary and sufficient) conditions on the sequence (an) under which the answers are affirmative. 2. Let X and Y be random variables in L2(Ω, F, P) such that E [X] = E [Y ] = E [XY ] = 0, E X 2 = E Y 2 = 1, and A be a sub-σ-algebra of F. (a) Show that E [E [X| A] E [Y | A]] ≤ 1 2 . (b) Assume in addition that X and Y are IID RVs. Find a sub-σ-algebra A of F such that E [E [X| A] E [Y | A]] = 1 2 . 3. Let (Xn) be IID RVs in L1 and denote by µ = E[X1] their common first moment. Let τ be a stopping time with E[τ ] < ∞ and Sn = Pn k=1 Xk. Show that Sτ ∈ L1 and compute E[Sτ ]. Computer Science Engineering & Technology C++ Programming HNJKBHKJBK BJMBN Share QuestionEmailCopy link Comments (0)