In a recent paper, we proposed a new estimation method for the
blind deconvolution of a linear system with discrete random input, when the
observations may be noise perturbed. We give here asymptotic properties of the
estimators in the parametric situation. With nonnoisy observations, the speed
of convergence is governed by the $l_1$-tail of the inverse filter, which may
have an exponential decrease. With noisy observations, the estimator satisfies
a limit theorem with known distribution, which allows for the construction of
confidence regions. To our knowledge, this is the first precise asymptotic
result in the noisy blind deconvolution problem with an unknown level of noise.
We also extend results concerning Hankel’s estimation to
Toeplitz’s estimation and prove a formula to compute Toeplitz forms that
may have interest in itself.