22 1. Preparatory material
Show that the converse statement (i.e., that (1.28) and (1.29) imply joint
independence) is true for k = 2, but fails for higher k. Can one find a correct
replacement for this converse for higher k?
Exercise 1.1.18.
(i) If X1,...,Xk are jointly independent random variables taking val-
ues in [0, +∞], show that
E
k
i=1
Xi =
k
i=1
EXi.
(ii) If X1,...,Xk are jointly independent absolutely integrable scalar
random variables taking values in [0, +∞], show that
k
i=1
Xi is
absolutely integrable, and
E
k
i=1
Xi =
k
i=1
EXi.
Remark 1.1.14. The above exercise combines well with Exercise 1.1.14.
For instance, if X1,...,Xk are jointly independent sub-Gaussian variables,
then from Exercises 1.1.14, 1.1.18 we see that
(1.30) E
k
i=1
etXi
=
k
i=1
EetXi
for any complex t. This identity is a key component of the exponential
moment method, which we will discuss in Section 2.1.
The following result is a key component of the second moment method.
Exercise 1.1.19 (Pairwise independence implies linearity of variance). If
X1,...,Xk are pairwise independent scalar random variables of finite mean
and variance, show that
Var(
k
i=1
Xi) =
k
i=1
Var(Xi)
and more generally,
Var(
k
i=1
ciXi) =
k
i=1
|ci|2Var(Xi)
for any scalars ci (compare with (1.12), (1.22)).
The product measure construction allows us to extend Lemma 1.1.7:
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