20 1. Preparatory material

From the theory of product measure, we have the following equivalent

formulation of joint independence:

Exercise 1.1.11. Let (Xα)α∈A be a family of random variables, with each

Xα taking values in a measurable space Rα.

(i) Show that the (Xα)α∈A are jointly independent if and only for every

collection of distinct elements α1,...,αk of A, and all measurable

subsets Ei ⊂ Rαi for 1 ≤ i ≤ k , one has

P(Xαi ∈ Ei for all 1 ≤ i ≤ k ) =

k

i=1

P(Xαi ∈ Ei).

(ii) Show that the necessary and suﬃcient condition (Xα)α∈A being k-

wise independent is the same, except that k is constrained to be

at most k.

In particular, a finite family (X1,...,Xk) of random variables Xi, 1 ≤ i ≤ k

taking values in measurable spaces Ri are jointly independent if and only if

P(Xi ∈ Ei for all 1 ≤ i ≤ k) =

k

i=1

P(Xi ∈ Ei)

for all measurable Ei ⊂ Ri.

If the Xα are discrete random variables, one can take the Ei to be

singleton sets in the above discussion.

From the above exercise we see that joint independence implies k-wise

independence for any k, and that joint independence is preserved under

permuting, relabeling, or eliminating some or all of the Xα. A single random

variable is automatically jointly independent, and so 1-wise independence

is vacuously true; pairwise independence is the first non-trivial notion of

independence in this hierarchy.

Example 1.1.13. Let F2 be the field of two elements, let V ⊂ F2

3

be

the subspace of triples (x1,x2,x3) ∈ F2

3

with x1 + x2 + x3 = 0, and let

(X1,X2,X3) be drawn uniformly at random from V . Then (X1,X2,X3)

are pairwise independent, but not jointly independent. In particular, X3 is

independent of each of X1,X2 separately, but is not independent of (X1,X2).

Exercise 1.1.12. This exercise generalises the above example. Let F be a

finite field, and let V be a subspace of

Fn

for some finite n. Let (X1,...,Xn)

be drawn uniformly at random from V . Suppose that V is not contained in

any coordinate hyperplane in

Fn.

(i) Show that each Xi, 1 ≤ i ≤ n is uniformly distributed in F.