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We then write

nn≥1

n

nn≥1

nn

nn≥1

The main objective of this book is to study certain families of stochastic (or random) processes in discrete time. There are two ways of seeing such objects:

 – as a sequence (Xn)n∈ℕ of real random variables;

 – as a single random variable X taking values in the set of real sequences.

nn∈ℕ

n

nnn∈ℕ

nn∈ℕ1nnnn∈ℕnn∈ℕ

nn∈ℕnn∈ℕn−1

nn−1

PROOF.– This proof directly follows from the definition of the objects. We have

and hence, the desired equality.

□

Indeed, this property completely characterizes the distribution of the process X according to the following theorem.

nn∈ℕnnnn∈ℕnnn∈ℕ

1 1) for any n ∈ ℕ, μn is defined on (En+1, ε⊗(n+1)),

2 2) for any n ∈ ℕ∗ and (A0,..., An−1) ∈ εn, we have μn−1(A0 × ... × An−1) = μn(A0 × ... × An−1 × E).

nn∈ℕnn∈ℕ

This 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).