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E. ∫ n ∫ Ψ ( x ) f X ( x ) dx < ∞ ) then gˆ ( y ) fY ( y ) dy < ∞ ). , Yn ) ; Random Vectors 55 5) linearity: Ε (λ1Ψ1 ( X ) + λ2 Ψ 2 ( X ) Y ) = λ1Ε ( Ψ1 ( X ) Y ) + λ2 Ε ( Ψ 2 ( X ) Y ) . The demonstrations which are generally easy may be found in the exercises. Let us note in particular that as far as the first property is concerned, it is sufficient to re-write the demonstration of the last proposition after stripping it of absolute values. The chapter on quadratic means estimation will make the notion of conditional expectation more concrete.

V. (density ⎛ 1 ⎛ x − m ⎞2 ⎞⎟ ⎟⎟ ⎟⎟ ). exp ⎜⎜⎜− ⎜⎜ ⎜⎝ 2 ⎝⎜ σ ⎟⎠ ⎠⎟⎟ 2πσ 1 It belongs to L1 (dP ) and L2 ( dP ) . v. (density 1 ). π (1 + x 2 ) It does not belong to L1 (dP ) and thus does not belong to L2 (dP ) either. 2. Properties – L1 (dP ) is a Banach space; we will not use this property for what follows; – L2 ( dP ) is a Hilbert space. We will give the properties here without any demonstration. * We can equip L2 ( dP ) with the scalar product defined by: ∀X , Y ∈ L2 (dP ) < X,Y > = ∫ X (ω )Y (ω ) dP (ω ) Ω This expression is well defined because following Schwarz’s inequality: Random Vectors ∫ 23 2 Ω X (ω )Y (ω ) dP (ω ) ≤ ∫ X 2 (ω ) dP (ω ) ∫ Y 2 (ω ) dP (ω ) < ∞ Ω Ω and the axioms of the scalar product are immediately verifiable.

We can also observe that Γ X −ΕX = Γ X . – Second-order complex random variables and vectors: we say that a complex random variable X = X1 + i X 2 is second order if X1 and X 2 ∈ L2 (dP ) . The covariance of two centered second-order random variables, X = X1 + i X 2 and Y = Y1 + iY2 has a natural definition: Cov ( X , Y ) = EXY = E ( X 1 + iX 2 )(Y1 − iY2 ) = E ( X 1Y1 + X 2Y2 ) + iE ( X 2Y1 − X 1Y2 ) and the decorrelation condition is thus: E ( X 1Y1 + X 2Y2 ) = E ( X 2Y1 − X 1Y2 ) = 0 . , n) X j = X 1 j + iX 2 j is a second-order complex random variable.