2 - Independence and conditioning

Summary

“Philosophy” of this chapter

1. (a) A probabilistic model {(Ω, ℱ, P); (X_i)_i∈I} consists of setting together in a mathematical way different sources of randomness, i.e. the r.v.s (X_i)_i∈I usually have some complicated joint distribution

It is always a simplification, and thus a progress, to replace this “linked” family by an “equivalent” family (Y_j)_j∈J of independent random variables, where by equivalence we mean the equality of their σ-fields: σ(X_i, i ∈ I)= σ(Y_j, j ∈ J) up to negligible sets.

1. (b) Assume that the set of indices I splits into I₁ + I₂, and that we know the outcomes {X_i(ω); i ∈ I₁}. This modifies deeply our perception of the randomness of the system, which is now reduced to understanding the conditional law of (X_i)_i∈I2, given (X_i)_i∈I1. This is the main theme of D. Williams' book [64].

2. (c) Again, it is of great interest, even after this conditioning with respect to (X_i)_i∈I1, to be able to replace the family (X_i)_i∈I2 by an “equivalent” family (Y_j)_j∈J2, which consists of independent variables, conditionally on (X_i)_i∈I1.

Note that the terms “independence” and “conditioning” come from our everyday language and are very suitable as translations of the corresponding probabilistic concepts. However, some of our exercises aim at pointing out some traps which may originate from this common language meaning.

1. (d) The Markov property (in a general framework) asserts the conditional independence of the “past” and “future” σ-fields given the “present” σ-field.