Download Homework 1. Due Oct 8, 2014

Homework 1. Due Oct 8, 2014
Q1. Let (Xn )n∈N be a sequence of i.i.d. real-valued random variables with µ := E[X1 ], and
P
φ(λ) := log E[eλX1 ] well-defined for all λ ∈ R. Let Sn := ni=1 Xi . Use exponential Markov
inequality to show that for any a > µ, there exists J(a) > 0 such that
P(Sn /n ≥ a) ≤ e−nJ(a)
∀ n ∈ N.
Similarly, for a < µ, there exists J(a) > 0 such that
P(Sn /n ≤ a) ≤ e−nJ(a)
∀ n ∈ N.
Use these two bounds to show that Sn /n → a almost surely as n → ∞.
Q2. Let (Xn )n∈N and X be R-valued random variables such that Xn → X a.s. Assume
1
1
further that kXn kp := E[|Xn |p ] p → kXkp := E[|X|p ] p < ∞ for some p > 0. Prove that
1
kXn − Xkp := E[|Xn − X|p ] p → 0.
Q3. Let (Xn )n∈N be a sequence of random variables defined on (Ω, F, P), such that Xn
converges in probability to a random variable X. Prove that we can find a subsequence
n1 < n2 < · · · , such that Xnk → X a.s. along this subsequence.
Q4. We have shown that if (Xn )n∈N are independent and uniformly bounded by some finite
P
P
constant C > 0, and n Xn converges a.s., then n V ar(Xn ) < ∞. By constructing an
example, show that if we only assume that V ar(Xn ) is uniformly bounded, then it may
P
P
happen that n Xn converges a.s., and yet n V ar(Xn ) = ∞.
Q5. Let X and Y be two independent random variables. Prove that if for some p > 0,
E[|X + Y |p ] < ∞, then E[|X|p ] < ∞ and E[|Y |p ] < ∞.
Q6. Let (ξn )n∈N be i.i.d. random variables taking values in the set of natural numbers N.
Assume that P(ξ1 = i) = pi > 0 for all i ∈ N. Let Dn denote the cardinality of the set
{ξ1 , . . . , ξn }. Prove that (i) Dn → ∞ a.s.; (ii) Dn /n → 0 in probability. Can one strengthen
(ii) to a.s. convergence?
Q7. Let (Xn )n∈N be a sequence of independent random variables. Let r be the radius of
P∞ Xn n
convergence of the power series
z . Prove that r is a non-random constant. If
n=1 e
(Xn )n∈N are furthermore identically distributed, then show that r = 0 if E[X1+ ] = ∞ and
r = 1 if E[X1+ ] < ∞.
Q8. Let (Xn )n∈N be a sequence of independent (but not necessarily identically distributed)
P
random variables., and let Sn := ni=1 Xi . Prove the following maximal inequality:
For all t > 0,
P( max |Si | ≥ t) ≤ 3 max P(|Si | ≥ t/3).
1≤i≤n
1≤i≤n
1