### Midterm Exam

```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}) ?
```