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.
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