34 1. Preparatory material

almost surely. Thus we see that the latter two notions are sensitive not only

to the distribution of the random variables, but how they are positioned in

the sample space.

The limit of a sequence converging almost surely or in probability is

clearly unique up to almost sure equivalence, whereas the limit of a sequence

converging in distribution is only unique up to equivalence in distribution.

Indeed, convergence in distribution is really a statement about the distri-

butions μXn , μX rather than of the random variables Xn,X themselves. In

particular, for convergence in distribution one does not care about how cor-

related or dependent the Xn are with respect to each other, or with X;

indeed, they could even live on different sample spaces Ωn, Ω and we would

still have a well-defined notion of convergence in distribution, even though

the other two notions cease to make sense (except when X is determin-

istic, in which case we can recover convergence in probability by Exercise

1.1.25(vi)).

Exercise 1.1.27 (Borel-Cantelli lemma). Suppose that Xn,X are random

variables such that

∑

n

P(d(Xn,X) ≥ ε) ∞ for every ε 0. Show that

Xn converges almost surely to X.

Exercise 1.1.28 (Convergence and moments). Let Xn be a sequence of

scalar random variables, and let X be another scalar random variable. Let

k, ε 0.

(i) If supn

E|Xn|k

∞, show that Xn has a tight sequence of distri-

butions.

(ii) If supn

E|Xn|k

∞ and Xn converges in distribution to X, show

that

E|X|k

≤ lim infn→∞

E|Xn|k.

(iii) If supn

E|Xn|k+ε

∞ and Xn converges in distribution to X, show

that

E|X|k

= limn→∞

E|Xn|k.

(iv) Give a counterexample to show that (iii) fails when ε = 0, even if

we upgrade convergence in distribution to almost sure convergence.

(v) If the Xn are uniformly bounded and real-valued, and

EXk

=

limn→∞ EXn

k

for every k = 0, 1, 2,..., then Xn converges in distri-

bution to X. (Hint: Use the Weierstrass approximation theorem.

Alternatively, use the analytic nature of the moment generating

function

EetX

and analytic continuation.)

(vi) If the Xn are uniformly bounded and complex-valued, and

EXkXl

= limn→∞

EXnXnl

k

for every k, l = 0, 1, 2,..., then Xn

converges in distribution to X. Give a counterexample to show

that the claim fails if one only considers the cases when l = 0.