Discrete Stochastic Processes and Optimal Filtering

6.7. Stability and convergence

Let us now study the stability and the convergence of the algorithm of the deterministic gradient.

By taking the recursive expressions of the coefficient vector and by translation:

The following expressions

enable us to write: α_K+1 = (I_d − 2μR)α_K Id: identity matrix.

By writing R in the form

and by premultiplying α_K+1 by Q⁻¹, we obtain:

Thus:

or:

and:

Thus, the algorithm is stable and convergent if

Thus, if and only if

Thus, if and only if:

In addition, we ...