5.1.1.– **(Module and phase joint law of a 2D Gaussian r.v.)** (see p. 20) As this substitution is bijective and its Jacobian is equal to *r*, which is positive, the joint distribution of the pair (*R*, Θ) has a density of:

Following property [1.3] we derive from *p*_{RΘ}(*r*, *θ*) = *g*(*r*)*h*(*θ*), where *g*(*r*) = and *h*(*θ*) = (2*π*)^{−1} (*θ* ∈ (0, 2*π*)), that the random variables *R* and Θ are independent. Therefore, Θ is uniform on (0, 2*π*) and *R* has a Rayleigh distribution.

5.1.2.– (*δ*-**method**) (see p. 21) We shall use the hypothesis cov (*X*_{0}, *X*_{1}) = *σ*^{2}*I*_{2}. The Jacobian, here noted *∂g*, of *g* : (*X*_{0}, *X*_{0}) → (*R*, *θ*) is deduced from the Jacobian *∂h* of *h* : (*R*, *θ*) → (*X*_{0}, *X*_{1}) following:

We have:

and, from [1.49]:

We deduce:

[5.1]

This result is therefore correct on the condition that ...

Start Free Trial

No credit card required