Advanced Probability EPFL - Fall Semester 2014-2015 Midterm Exam SURNAME: ................................................ FIRST NAME: ........................................................... Exercise 1. For this exercise, we introduce the following useful notations: Hn = n X 1 j=1 j and Kn = n X 1 j2 j=1 and recall the asymptotic results Hn =1 n→∞ log(n) lim and lim Kn = n→∞ π2 6 Let now (Xn , n ≥ 1) be a sequence of independent random variables defined on the same probability space (Ω, F, P). Assume that P({Xn = n}) = n12 and P({Xn = 0}) = 1 − n12 , for n ≥ 1. Let also Sn = X1 + . . . + Xn . a) Compute E(Sn ) and Var(Sn ). b) Does there exist a constant c1 ∈ R such that If yes, prove it; if no, explain why. Sn n converges in probability to c1 as n → ∞? Sn c) Does there exist a constant c2 ∈ R such that nH converges almost surely to c2 as n → ∞? n If yes, prove it; if no, explain why. P Hint: It holds that n≥2 n (log1 n)2 < ∞. BONUS: prove it! d) To what limit does P({Sn = 1}) converge as n → ∞? Exercise 2. Let (Xn , n ≥ 1) be a sequence of i.i.d. random variables defined on the same probability space (Ω, F, P). Assume that X1 ∼ N (0, 1). Let also Yn = Xn2 and Sn = Y1 + . . . + Yn , for every n ≥ 1. Here are some facts which might be useful for this exercise: - If X ∼ N (0, 1), then E(X 4 ) = 3, P({|X| ≤ 1}) ' 0.68, P({|X| ≤ 2}) ' 0.95. r R π 2 . - If a > 0, then R exp(−ax ) dx = a a) Use the central limit theorem to estimate the following probability: P({S200 ≥ 220}) 1 b) Compute the value of Λ(s) = log(E(exp(sY1 ))) for s < . 2 (NB: It can be shown that for s ≥ 12 , Λ(s) = +∞.) c) Compute Λ∗ (t) = sups∈R (st − Λ(s)) for t > E(Y1 ). d) Deduce an upper bound on P({Sn > nt}) for t > E(Y1 ). e) Numerical application: what does your upper bound give for P({S200 ≥ 220}) ?