result is very important for the theory of processes as it signifies that it is sufficient to specify (all) the finite-dimensional distributions and for them to be compatible with each other, to uniquely define a process distribution over the space of infinite random sequences. In practice, this makes it possible to justify the construction of processes (existence property) as well as showing that two processes have the same distribution (unicity property).
To study the random variables taking values in the set of sequences, we need new definitions for σ-algebras and measurability.
DEFINITION 1.20.– In a probability space (Ω,
When (
EXAMPLE 1.23.– Let (Xn)n∈ℕ be a sequence of random variables and we consider, for any n ∈ ℕ,
DEFINITION 1.21.– Let (Ω,
– X is said to be adapted to the filtration (n)n∈ℕ (or again (n)n∈ℕ−adapted), if Xn is n-measurable for any n ∈ ℕ;
– X is said to be predictable with respect to the filtration (n)n∈ℕ (or again (n)n∈ℕ−predictable), if Xn is n−1-measurable for any n ∈ ℕ∗.
EXAMPLE 1.24.– A process is always adapted with respect to its natural filtration.
As its name indicates, for a predictable process, we know its value Xn from the instant n − 1.
1.4. Exercises
EXERCISE 1.1.– Let Ω = {a, b, c}.
1 1) Completely describe all the σ-algebras of Ω.
2 2) State which are the sub-σ-algebras of which.
EXERCISE 1.2.– Let Ω = {a, b, c, d}. Among the following sets, which are σ-algebras?
1 1)
2 2)
3 3)
4 4)
For those which are not σ-algebras, completely describe the σ-algebras they generate.
EXERCISE 1.3.– Let X be a random variable on (Ω,
EXERCISE 1.4.– Let A ∈
EXERCISE 1.5.– Let Ω = {P, F} × {P, F} and =
– X1 be the random variable number of T on the first toss;
– X2 be the number of T on the second toss;
– Y be the number of T obtained on the two tosses;
– and Z = 1 if the two tosses yielded an identical result; otherwise, it is 0.
1 1) Describe 1 = σ(X1) and 2 = σ(X2). Is X1 2-measurable?
2 2) Describe = σ(Y). Is Y 1-measurable? Is X1 -measurable?
3 3) Describe = σ(Z). Is Z 1-measurable, -measurable? Is X1 -measurable?
4 4) Give the inclusions between , 1, 2, and .
EXERCISE
