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: