Chapter 1

Random Vectors

1.1. Definitions and general properties

If we remember that n = {x = (x1,.,xn)| xj ;j=1 to n}, the set of real n -tuples can be fitted to two laws: and making it a vector space of dimension n.

The basis implicitly considered on n will be the canonical base l1 = (1, 0,., 0),., ln = (0,., 0, 1) and x n expressed in this base will be denoted:

Definition of a real random vector

Beginning with a basic definition, without concerning ourselves at the moment with its rigor: we can say simply that a real vector linked to a physical or biological phenomenon is random if the value taken by this vector is unknown and the phenomenon is not completed.

For typographical reasons, the vector will instead be written XT = (X1,., Xn) or even X = (X1,., Xn) when there is no risk of confusion.

In other words, given a random vector X and ? ? n we do not know if the assertion (also called the event) (X ?) is true or false:

However, we do usually know the "chance" that X ?; this is denoted P (X B) and is called the probability of the event (X B).

After completion of the phenomenon, the result (also called the realization) will be denoted

when there is no risk of confusion.

An exact definition of a real random vector of dimension n will now be given. We take as given that:

- O = basic space. This is the set of all possible results (or tests) ? linked to a random phenomenon.

- a = s -algebra (of events) on O, recalling the axioms:

1) O a

2) if A a then the complementary Ac a

3) if (?j, j J) is a countable family of events then is an event, i.e.

- n = space of observables

- (n) = Borel algebra on n; this is the smallest s -algebra on n which contains all the open sets of n.

DEFINITION.- X is said to be a real random vector of dimension n defined on (O, a) if X is a measurable mapping (O,a)(n,(n)), i.e. ?B(n) X-1(B)a.

When n = 1 we talk about a random variable (r.v.).

In the following the event X-1(B) is also denoted as {?|X(?) B} and even more simply as (X B).

PROPOSITION.- In order for X to be a real random vector of dimension n (i.e. a measurable mapping (O,a)(n,(n)), is necessary and it suffices that each component Xj j = 1 at n is a real r.v. (i.e. is a measurable mapping (O,a)(,())).

ABRIDGED DEMONSTRATION.- It suffices to consider:

as we show that where denotes the s-algebra generated by the measurable blocks B1 ×.× ?n.

Now , which concerns a if and only if each term concerns a, that is to say if each Xj is a real r.v.

DEFINITION.- X = X1 + iX2 is said to be a complex random variable defined on (O, a) if the real and imaginary parts X1 and X2 are real variables, that is to say the random variables X1 and X2 are measurable mappings (O, a) (, B()).

EXAMPLE.- The complex r.v. can be associated with a real random vector X = (X1,., Xn) and a real n -tuple, u = (u1,., un) n.

The study of this random variable will be taken up again when we define the characteristic functions.

Law

Law PX of the random vector X.

First of all we assume that the s-algebra a is provided with a measure P, i.e. a mapping P: a [0, 1] verifying:

1) P (O) = 1

2) For every family (Aj, j J) of countable pairwise disjoint events:

DEFINITION.- We call the law of random vector X, the "image measure PX of P through the mapping of X", i.e. the definite measure on (n) defined in the following way by: ?B(n)

Terms 1 and 2 on the one hand and terms 3, 4 and 5 on the other are different notations of the same mathematical notion.

Figure 1.1. Measurable mapping X

It is important to observe that as the measure P is given along a, PX(B) is calculable for all B(n) because X is measurable.

The space n provided with the Borel algebra (n) and then with the PX law is denoted: (n,(n),PX).

NOTE.- As far as the basic and the exact definitions are concerned, the basic definition of random vectors is obviously a lot simpler and more intuitive and can happily be used in basic applications of probability calculations.

On the other hand in more theoretical or sophisticated studies and notably in those calling into play several random vectors, X, Y, Z,. considering the latter as definite mappings on the same space (O, a),

will often prove to be useful even indispensable.

Figure 1.2. Family of measurable mappings

In effect, the expressions and calculations calling into play several (or the entirety) of these vectors can be written without ambiguity using the space (O, a, P). Precisely, the events linked to X, Y, Z,. are among elements A of a (and the probabilities of these events are measured by P).

Let us give two examples:

1) if there are 2 random vectors X,Y:(O,a,P)(n,(n)) and given B and B´(n), the event (X B) n (Y B´) (for example) can be translated by X-1 (B) n Y-1 (B´) a;

2) there are 3 r.v. X,Y,Z:(O,a,P)(,()) and given .

Let us try to express the event (Z = a - X - Y).

Let us state U = X, Y, Z and B = {(x, y, z) 3 | x + y + z = a}.

where B Borel set of 3, represents the half-space bounded by the plane (?) not containing the origin 0 and based on the triangle A B C.

Figure 1.3. Example of Borel set of 3

U is (O,a)(3,(3)) measurable and:

NOTE ON SPACE (O, a,P).- We said that if we took as given O and then a on O and then P on a and so on, we would consider the vectors X, Y, Z,. as measurable mappings:

This way of introducing the different concepts is the easiest to understand, but it rarely corresponds to real probability problems.

In general (O, a, P) is not specified or is even given before "X, Y, Z,. measurable mappings". On the contrary, given the random physical or biological sizes X, Y, Z,. of n, it is on departing from the latter that (O, a, P) and X, Y, Z,. definite measurable mappings on (O, a, P) are simultaneously introduced. (O, a, P) is an artificial space intended to serve as a link between X, Y, Z,.

What has just been set out may seem exceedingly abstract but fortunately the general random vectors as they have just been defined are rarely used in practice.

In any case, and as far as we are concerned, we will only have to manipulate in what follows the far more specific and concrete notion of a "random vector with a density function".

DEFINITION.- We say that the law PX of the random vector X has a density if there is a mapping: which is positive and measurable, called the density of PX such that ?B(n).

VOCABULARY.- Sometimes we write

and we say also that the measure PX admits the density fX with respect to the Lebesgue measure on n. We also say that the random vector X admits the density fX.

NOTE.- ?B fX(x1.xn)dx1.dxn = P(Xn)=1.

For example, let the random vector be X = (X1, X2, X3) of density fX (x1, x2, x3) = K x3 1? (x1, x2, x3) where ? is the half sphere defined by with x3 = 0.

We easily obtain via a passage through spherical...