*3.2. Asymptotic Properties*

In this section, we establish the consistency and the asymptotic distributions for the estimators *θ* 4*c <sup>ϕ</sup>* and 4*t c θ* 4*c ϕ* . In order to prove the consistency of the estimators, we adopted the results from the general theory of Z-estimators as presented for example in [28]. Then, using the consistency of the estimators, as well as supplementary conditions, we proved that the asymptotic distributions of the estimators are multivariate normal:

#### **Assumption 1.**

*(a) There exist compact neighbourhoods Vθ*<sup>0</sup> *of θ*<sup>0</sup> *and Vt<sup>θ</sup>*<sup>0</sup> *of tθ*<sup>0</sup> *such that*

$$\int \sup\_{\theta \in V\_{\theta\_0}, t \in V\_{t\_0}} ||\Psi(y, \theta, t)|| dP\_0(y) < \infty.$$

*(b) For any positive ε, the following condition holds*

$$\inf\_{\left(\vartheta,t\right)\in M} \parallel \int \Psi\left(y,\theta,t\right) \parallel dP\_0\left(y\right) > 0 = \parallel \int \Psi\left(y,\theta\_0,t\_{\theta\_0}\right) dP\_0\left(y\right) \parallel \cdot$$

*where M* := {(*θ*, *t*) s.t. (*θ*, *t*) − (*θ*0, *tθ*<sup>0</sup> ) > *ε*}*.*

**Proposition 1.** *Under Assumption 1, θ* 4*c <sup>ϕ</sup> converges in probability to θ*<sup>0</sup> *and* 4*t c θ* 4*c ϕ converges in probability to tθ*<sup>0</sup> *:*
