1
Elementary Probabilities and an Introduction to Stochastic Processes

This chapter reviews the basic concepts related to probability and random variables which will be useful for the rest of this text. For a more detailed explanation as well as demonstrations, the readers may refer to [BAR 07, DAC 82, FOA 03, OUV 08, OUV 09] in French and [BIL 12, CHU 01, DUR 10, KAL 02, SHI 00] in English. The readers who are already familiar with these concepts may proceed straight to section 1.3, which introduces the concept of stochastic processes.

This chapter begins with a brief summary of the concepts of a s-algebra in section 1.1. These concepts will help in understanding the construction of the properties of conditional expectation in Chapter 2. We then study the chief definitions and properties of random variables and their distribution in section 1.2. There is an emphasis on discrete random variables as this entire book essentially studies discrete cases. Section 1.3 defines a stochastic process, which is the main subject studied in this book. Finally, there are exercises in handling these different concepts in section 1.4. The solutions are given in Chapter 8.

Throughout the rest of the text, O is a non-empty set and (O) denotes the set of the subsets of O :

The set O is called the universe or the fundamental set. In practice, the set O contains all the possible outcomes of a random experiment.

1.1. Measures and s-algebras

Let us start by reviewing the concept of a s-algebra.

DEFINITION 1.1.- A subset of (O) is a s-algebra over O if

1. 1) O ? ;
2. 2) is stable by complementarity: for any A ? , we have Ac ? , where Ac denotes the complement of A in O: Ac = O\A;
3. 3) is stable under a countable union: for any sequence of elements (An)n?N of , we have

Elements of a s-algebra are called events.

EXAMPLE 1.1.- The set = {Ø, O} is a s-algebra and is also the smallest s-algebra over O; it is called the trivial s-algebra. Indeed, is in fact a s-algebra since O ? and by creating unions of Ø and O we always obtain Ø ? or O ? . Further, for any other s-algebra , we clearly have ? .

EXAMPLE 1.2.- The set (O) is the largest s-algebra over O; it is called the largest s-algebra. Indeed, by construction, (O) contains all the subsets of O, and thus it contains in particular O and it is stable by complementarity and under countable unions. In addition, any other s-algebra over O is clearly included in (O).

DEFINITION 1.2.- Let O be a non-empty set and be a s-algebra over O. The couple (O, ) is called a probabilizable space.

Among the elementary properties of s-algebra, we can cite stability through any intersection (countable or not).

PROPOSITION 1.1.- Any intersection of s-algebras over a set O is a s-algebra over O.

PROOF.- Let (i)i?I be any family of s-algebra indexed by a non-empty set I. Thus,

• - first of all, for any i, O ? i, thus O ? ni?Ii;
• - secondly, if A ? ni?I i, then for any i, A ? i. As these are s-algebras, we have that for any i, Ac ? i, thus Ac ? ni?I i;
• - finally, if for any n ? N, An ? ni?I i, then for any i, n, An ? i. As these are s-algebras, we have that for any i, ?n?NAn ? i, thus

It is generally difficult to make explicit all the events in a s-algebra. We often describe it using generating events.

DEFINITION 1.3.- Let e be a subset of (O). The s-algebra s(e) generated by e is the intersection of all s-algebras containing e. It is the smallest s-algebra containing e. e is called the generating system of the s-algebra s(e).

It can be seen that s(e) is indeed a s-algebra, being an intersection of s-algebras.

EXAMPLE 1.3.- If A ? O, then, s(A) = {Ø, O, A, Ac} is the smallest s-algebra O containing A.

EXAMPLE 1.4.- If O is a topological space, the s-algebra generated by the open sets of O is called the Borel s-algebra of O. A Borel set is a set belonging to the Borel s-algebra. On R, (R) generally denotes the s-algebra of Borel sets. It must be recalled that this is also the s-algebra generated by the intervals, or by the intervals of the form ] - 8, x], x ? R. Thus, there is no unicity of the generating system.

We will now recall the concept of the product s-algebra.

DEFINITION 1.4.- Let (Ei, i)i?N be a sequence of measurable spaces.

• - Let n ? N. The s-algebra defined over and generated by

is denoted by 0 ? ... ? n, and it is called the product s-algebra over We have, in particular,

In the specific case where E0 = ... = En = E and 0 = ... = n = , we also write

• - We use ?i?Ni to denote the s-algebra over the countable product space generated by the sets of the form where Ai ? i and Ai = Ei except for a finite number of indices i. In the specific case where, for any and i = , the product space is denoted by EN, and the s-algebra ?i?Ni is denoted by ?N.

Finally, let us review the concepts of measurability and measure.

DEFINITION 1.5.- Let O be non-empty set and be a s-algebra on O.

• - A measure over a probabilizable space (O, ) is defined as any mapping µ defined over , with values in [0, +8] = R+ ? {+8}, such that µ(Ø) = 0 and for any family (Ai)i?N of pairwise disjoint elements of , we have the property of s-additivity:
• - A measure µ over a probabilizable space (O, ) is said to be finite, or have finite total mass, if µ(O) < 8.
• - If µ is a measure over a probabilizable space (O, ), then the triplet (O, , µ) is called a measured space.

DEFINITION 1.6.- Let (O, ) and (E, e ) be two probabilizable spaces. A mapping X, defined over O taking values in E, is said to be (, e)-measurable, or just measurable, if there is no ambiguity regarding the reference s-algebras, if

In practice, when E ? R, we set e = (E) the set of Borel subsets of E, that is, the set of subsets of E. We can simply say that X is -measurable. When, in addition, we manipulate a single s-algebra over O, it can be simply said that X is measurable. If we work with several s-algebras over O, the concerned s-algebra must always be specified: X is -measurable.

EXAMPLE 1.5.- If (O, ) is a measurable space and A ? , then the indicator function

is -measurable. Indeed, for any Borel set B in R, we have

Thus, in all cases, we do have

EXAMPLE 1.6.- The composition of two measurable functions is measurable. Indeed, if (O, ), (E, e) and (G, ) are three probabilizable spaces, f : O ? E and g : E ? G are two (, e) and (e, )-measurable mappings, respectively, then for any B ? , g-1(B) ? e and consequently,

Thus, the composition g ° f is indeed measurable on (O, ) in (G, ).

1.2. Probability...