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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.

On the random functional central limit theorems with almost sure convergence

Orzóg M., Rychlik Z.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 1

125--138

In this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems.

An almost sure limit theorem for the maxima and sums of stationary Gaussian sequences

Dudziński M.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

139--152

Let X1, X2,… be some standardized stationary Gaussian process and let us put: Mk = max(X1,…Xk), Sk = [wzór] Xi, σk = [wzór]. Our purpose is to prove an almost sure central limit theorem for the sequence (Mk, Sk/σk) under suitable normalization of Mk. The investigations presented in this paper extend the recent research of Csaki and Gonchigdanzan [1] and Dudziński [2].