Weighted Limit Theorems for Continuous-Time Vector Martingales with Explosive and Mixed Growth

Abstract

In this article, we establish the almost-sure central limit theorem (ASCLT) for a quasi-left continuous vector martingale with explosive and mixed (regular and explosive) growth. We also prove a quadratic extension and establish several new central limit theorems associated with the obtained ASCLT. Finally, we study the problem of parameter estimation in the particular case of multidimensional diffusion processes, which illustrates in a concrete manner the use of our results.

Asymptotic Properties of the LS-estimator of a Gaussian Autoregressive Process by an Averaging Method

“…Using the stable QSL property, we construct an estimator˜ 2 of the noise covariance 2 and we specify its weak convergence rate. In the unstable case, using the limit theorems for continous-time martingales with explosive growth (see Fathallah and Kebaier, 2012), we establish the same asymptotic properties of the LS-estimatorˆ of with arithmetic convergence rates. We use the unstable QSL property, we construct an estimatorˆ 2 of the noise covariance 2 and we specify its weak convergence rate.…”

“…Applying the QSL (see Theorem 2.1.1 in Fathallah and Kebaier, 2012) to the couple M V where V = V t = e B t t ≥ 0 , we obtain the third assertion of lemma 3.4: …”

“…Applying the ASCLT (see Theorem 2.1.1 in Fathallah and Kebaier, 2012) to the couple M V where M is a continuous vector martingale with explosive growth given by relation (57) …”