Analysis of the Algorithm in the General Case

Abstract

In the stochastic algorithms studied in Chapter 1 there is a fairly strong condition on the moments of the process (X_n)_n≥0 (Assumption (A.5)): for any compact set Q, there exists a constant μ_q(Q) < ∞ such that for any initial condition (x, a)

$${E_{{x,a}}}\left\{ {I\left( {{\theta_k} \in Q,k \leqslant n} \right)\left( {1 + {{\left| {{X_{{n + 1}}}} \right|}^q}} \right)} \right\} \leqslant {\mu_q}(Q)\left( {1 + {{\left| x \right|}^q}} \right)$$

As soon as the evolution of the process (X_n) depends on the sequence (θ_n) (unlike the equaliser discussed in Chapter 2), as in the algorithm with general linear dynamics, there is no reason for (A.5) to be satisfied.