A projected stochastic approximation method for adaptive filters and identifiers

Generally, when stochastic approximation is used to identify the coefficients of a linear system or for an adaptive filter or equalizer, the iterate X
n
is projected back onto some finite set
G={x:|x_{i}|\leq B
, all
i
}, if it ever leaves it. The convergence of such truncated sequences have been discussed informally. Here it is shown, under very broad conditions on the noises, that
\{X_{n}\}
converges with probability 1 to the closest point in
G
to the optimum value of X
n
. Also, under even weaker conditions, the case of constant coefficient sequence is treated and a weak convergence result obtained. The set
G
is used for simplicity. It can be seen that the result holds true in more general cases, but the box is used since it is the only commonly used constraint set for this problem.