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The couple (X, Y) is, therefore, a random vector (referred to here as a “random couple”), with joint distribution defined by

for any (i, j) X(Ω) × Y (Ω), which makes it possible to derive the distributions of X and Y (called the marginal distributions of the couple (X, Y )):

Distribution of X:

Distribution of Y :

To conclude this section on random variables, we will review some classic results of convergence for sequences of random variables. Throughout the rest of this book, the abbreviation r.v. signifies random variable.

nn≥1

1 1) It is assumed that there exists p > 0 such that, for any n ≥ 0, [|Xn|p] < ∞, and [|X|p] < ∞. It is said that the sequence of random variables (Xn)n≥1 converges on the average of the order p or converges in Lp towards X, ifWe then write In the specific case where p = 2, we say there is a convergence in quadratic mean.

2 2) The sequence of r.v. (Xn)n≥1 is called almost surely (a.s.) convergent towards X, if