10 1. Preparatory material

not consider in this course matrices whose shapes are themselves

random variables. One can view a matrix-valued random variable

X = (Xij)1≤i≤n;1≤j≤p as the joint random variable of its scalar com-

ponents Xij. One can apply all the usual matrix operations (e.g.,

sum, product, determinant, trace, inverse, etc.) on random matri-

ces to get a random variable with the appropriate range, though

in some cases (e.g., with inverse) one has to adjoin the undefined

symbol ⊥ as mentioned earlier.

(ix) Point processes, which take values in the space N(S) of subsets A

of a space S (or more precisely, on the space of multisets of S, or

even more precisely still as integer-valued locally finite measures

on S), with the σ-algebra being generated by the counting func-

tions |A ∩ B| for all precompact measurable sets B. Thus, if X

is a point process in S, and B is a precompact measurable set,

then the counting function |X ∩ B| is a discrete random variable

in {0, 1, 2,...} ∪ {+∞}. For us, the key example of a point pro-

cess comes from taking the spectrum {λ1,...,λn} of eigenvalues

(counting multiplicity) of a random n × n matrix Mn. Point pro-

cesses are discussed further in [Ta2010b, §2.6]. We will return to

point processes (and define them more formally) later in this text.

Remark 1.1.6. A pedantic point: strictly speaking, one has to include

the range R = (R, R) of a random variable X as part of that variable

(thus one should really be referring to the pair (X, R) rather than X). This

leads to the annoying conclusion that, technically, Boolean random variables

are not integer-valued, integer-valued random variables are not real-valued,

and real-valued random variables are not complex-valued. To avoid this

issue we shall abuse notation very slightly and identify any random variable

X = (X, R) to any coextension (X, R ) of that random variable to a larger

range space R ⊃ R (assuming of course that the σ-algebras are compatible).

Thus, for instance, a real-valued random variable which happens to only

take a countable number of values will now be considered a discrete random

variable also.

Given a random variable X taking values in some range R, we define

the distribution μX

of X to be the probability measure on the measurable

space R = (R, R) defined by the formula

(1.2) μX(S) := P(X ∈ S),

thus μX is the pushforward X∗P of the sample space probability measure

P by X. This is easily seen to be a probability measure, and is also a

probabilistic concept. The probability measure μX is also known as the law

for X.