Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Let {Zn} be a real nonstationary stochastic process such that E(Zn|Fn.1)a.s.<. and E(Z2n|Fn.1)a.s.<., where {Fn} is an increasing sequence of .-algebras. Assuming that E(Zn|Fn.1)=gn(.0,.0)=g(1)n(.0)+g(2)n(.0,.0), .0..p, p<., .0..q and q.., we study the asymptotic properties of ..n:=argmin..nk=1(Zk.gk(.,..))2..1k, where .k is Fk.1-measurable, ..={..k} is a sequence of estimations of .0, gn(., ..) is Lipschitz in . and gn(2)(.0, ..).gn(2)(., ..) is asymptotically negligible relative to gn(1)(.0).gn(1)(.). We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of {..n} in the linear model. For that, we prove a strong law of large numbers for a class of submartingales. Again using this strong law, we derive the general conditions leading to the asymptotic distribution of ..n. We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.