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We investigate an almost sure limit theorem (ASCLT) for sequences of randomvariables having the form of a ratio of two terms such that the numeratorsatisfies the ASCLT and the denominator is a positive term which convergesalmost surely to 1. This result leads to the ASCLT for least square estimatorsfor Ornstein-Uhlenbeck process driven by fractional Brownian motion.

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