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Realized statistics based on high frequency returns have become very popular in financial economics. In recent years, different non-parametric estimators of the variation of a log-price process have appeared. Among them are the realized quadratic (co-)variation which is perhaps the most well known example, providing a consistent estimator of the integrated (co-)volatility when the logarithmic price process is continuous. In this paper, we propose to study the large deviation properties of realized (co-)volatility. Our main motivation is to improve upon the existing limit theorems such as the weak law of large numbers or the central limit theorem which have been proved in different contexts. Our large deviations results can be used to evaluate and approximate tail probabilities of realized (co-)volatility. As an application we provide the large deviations for the standard dependence measures between the two assets returns such as the realized regression coefficients or the realized correlation. Our study should contribute to the recent trend of research on the (co-)variance estimation problems, which are quite often discussed in high-frequency financial data analysis.

Proofs and Technical Appendix for paper: Spectral characterization of the non-independent increment family of Alpha-Stable processes that generalize Gaussian process models.

The characterization of spatial or temporal processes by a family of sufficient functions such as via a unique spectral representation and a mean function forms the basis of a large number of statistical modelling approaches. For instance, in Gaussian process regression modelling, the mean and covariance function specify uniquely the properties of the resulting statistical model. One can therefore parameterize such regression models and understand their structure and attributes generally via a specification of the covariance kernel. In this paper we generalize significantly the class of available spatial and temporal processes that may be used in statistical applications like regressions to allow for non-stationarity and heavy-tailedness. Importantly, we are able to demonstrate for the first time how to achieve this through a novel formulation no longer requiring independence of the increments in the stochastic construction of the process and statistical model. Furthermore, we achieve this in a manner akin to the covariance kernel specification in a Gaussian process model, we develop a novel covariation spectral representation of some non-stationary and non-indepenent increments symmetric Alpha-stable processes (SalphaS). Such a representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence and should allow a very wide class of statistical regression models to be subsequently developed. We present a general framework for sufficient conditions to characterize such processes and develop general constructive approaches to building models satisfying these conditions.

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in [D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (3) (2008) 727–759. Available on Mathematics arXiv:math.PR/0703355, 2007]. This approach allows us to recover and extend in a unified way some known criteria in the euclidean case (Bakry and Emery, Wang, Kusuoka and Stroock, …).

We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov–Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker–Planck equation recently studied by Hérau and Nier, Helffer and Nier, and Villani is in particular discussed in the final section.

In this paper we study quadratic transportation cost inequalities. To this end we introduce new families of inequalities (for quadratic transportation cost and for relative entropy) that are shown to be equivalent to the Poincare inequality. This allows us to give some examples of measures satisfying T 2 but not the logarithmic-Sobolev inequality.

A moderate deviation principle for non-linear functionals, with at most quadratic growth, of moving average processes (or linear processes) is established. The main assumptions on the moving average process are a Logarithmic Sobolev Inequality for the driving random variables and the continuity, or some (weaker) integrability condition on the spectral density (covering some cases of long range dependence). We also obtain the moderate deviation estimate for the empirical periodogram, exhibiting an interesting new form of the rate function, i.e. with a correction term compared to the Gaussian rate functional. As statistical applications we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter of a stationary autoregressive process and of the Neyman–Pearson likelihood ratio test in the Gaussian case.

We study exponentially asymptotic behaviors for the trajectories of additive functionals of Harris Markov chains. In the main result, we establish a moderate deviation principle for a pair of additive functionals of different growth rates. Moreover, we give explicit formulas for the rate functions which exhibit a non-quadratic behavior. In particular, we achieve the functional moderate deviations in two different scales. As an application, we obtain a functional law of LIL, which leads to a variety of strong limit laws in the spirit of Strassen [Z. Wahr. Geb. 3 (1964) 211–226].

Resume Nous demontrons ici un principe de deviations moderees (PDM) pour le principe de moyennisation d'une equation differentielle stochastique (EDS), avec une petite diffusion, plongee dans un milieu aleatoire rapide modelise par un processus de Markov exponentiellement ergodique et independant du mouvement Brownien conduisant cette EDS. Notre preuve repose sur la methode de Puhalskii pour la tension exponentielle et un PDM pour des fonctionnelles inhomogenes de processus de Markov etablis dans [5].

Resume Nous nous interessons dans cette Note aux deviations moderees pour le processus empirique fonctionnel d'une chaine de Markov a valeurs dans un espace d'etat general, possedant un atome. Ce principe est etabli sous les hypotheses suivantes : une condition sur le premier temps d'entree dans l'atome, et des conditions sur la classe de fonctions. Le cas general est egalement presente sous des conditions sur la “split chain” associee. Ces resultats peuvent etre consideres comme l'extension aux chaines de Markov de la caracterisation des deviations moderees de v.a. i.i.d. de Ledoux.