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Tangential existence and comparison, with applications to single and multiple integration

Kallenberg O.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 1

21--52

Two semi-martingales with respect to a common filtration are said to be tangential if they have the same local characteristics. When the latter are non-random, the underlying semi-martingale is known to have independent increments. We show that every semi-martingale has a tangential process with conditionally independent increments. We also extend the Zinn-Hitchenko and related tangential comparison theorems to continuous time. Combining those results, we obtain some surprisingly general existence, convergence, and tightness criteria for broad classes of single and multiple stochastic integrals.

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.

Maejima M., Tudor C. A.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

167--186

We study selfsimilar processes with stationary increments in the second Wiener chaos. We show that, in contrast with the first Wiener chaos which contains only one such process (the fractional Brownian motion), there is an infinity of selfsimilar processes with stationary increments living in the Wiener chaos of order two. We prove some limit theorems which provide a mechanism to construct such processes.

On the rate of convergence in non-central asymptotics of the Hermite variations of fractional Brownian sheet

Breton J. -C

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

301--311

The Hermite variations of the anisotropic fractional Brownian sheet enjoy similar behaviour to that for the fractional Brownian motion: central (convergence to a normal distribution) or non-central (convergence to a Hermite-type distribution). In this note, we investigate the rate of convergence in the non-central case.

Convergence in variation of the joint laws of multiple stable stochastic integrals

Breton J. -C

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

21--40

In this note, we are interested in the regularity in the sense of total variation of the joint laws of multiple stable stochastic integrals. Namely, we show that the convergence [formula] holds true as long as each kernel finconverges when n→+∞to fi in the Lorentz-type space [formula]. This result generalizes [4] from the one-dimensional case to the joint law case. It generalizes also [6] from the Wiener–Itô setting to the stable setting and [5] in the study of joint law of multiple stable integrals.