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.
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