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Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein-Uhlenbeck processes

Cénac P., Es-Sebaiy K.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

285--300

We will investigate an almost sure central limit theorem (ASCLT) for sequences of random variables having the form of a ratio of two terms such that the numerator satisfies the ASCLT and the denominator is a positive term which converges almost surely to one. This result leads to the ASCLT for least squares estimators for Ornstein-Uhlenbeck proces driven by fractional Brownian motion.

Almost sure properties of weighted vectorial martingales transforms with applications to prediction for linear regression models

Cénac P.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

61--76

We establish new almost sure properties for powers of weighted martingale transforms. It allows us to deduce useful asymptotic results for cumulative prediction and estimation errors associated with linear regression models. We also provide two examples of applications on the linear and functional autoregressive models.