arXiv subjects

Ehsan Azmoodeh

Publications and source records attributed to Ehsan Azmoodeh.

33 records · Page 2Linked to original sources

arXiv2016

Generalization of the Nualart-Peccati criterion

The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable $N$ of a given sequence $\{X_n\}_{n\ge1}$ of multiple Wiener-Itô integrals of fixed order, if $\mathbb {E}[X_n^2]\to1$ and $\mathbb {E}[X_n^4]\to \mathbb {E}[N^4]=3$. Since its appearance in 2005, the natural question of ascertaining which other moments can replace the fourth moment in the above criterion has remained entirely open. Based on the technique recently introduced in [J. Funct. Anal. 266 (2014) 2341-2359], we settle this problem and establish that the convergence of any even moment, greater than four, to the corresponding moment of the standard Gaussian distribution, guarantees the central convergence. As a by-product, we provide many new moment inequalities for multiple Wiener-Itô integrals. For instance, if $X$ is a normalized multiple Wiener-Itô integral of order greater than one, \[\forall k\ge2,\qquad \mathbb {E}\bigl[X^{2k}\bigr]>\mathbb {E} \bigl[N^{2k}\bigr]=(2k-1)!!.\]

math.PR↗

arXiv2016

Stein's method on the second Wiener chaos : 2-Wasserstein distance

In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called Stein equations at the heart of Stein's method. We provide several applications, and discuss comparison with recent similar results on the same topic.

math.PR↗

arXiv2015

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian--fractional Brownian model

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter $H$ of the fractional part satisfies $H\in(3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in(1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in(1/2,3/4)$, the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator.

math.PR↗

arXiv2015

Optimal Berry-Esseen bounds on the Poisson space

We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Biermé, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs.

math.PR↗

arXiv2015

A general approach to small deviation via concentration of measures

We provide a general approach to obtain upper bounds for small deviations $ \mathbb{P}(\Vert y \Vert \le ε)$ in different norms, namely the supremum and $β$- Hölder norms. The large class of processes $y$ under consideration takes the form $y_t= X_t + \int_0^t a_s d s$, where $X$ and $a$ are two possibly dependent stochastic processes. Our approach provides an upper bound for small deviations whenever upper bounds for the \textit{concentration of measures} of $L^p$- norm of random vectors built from increments of the process $X$ and \textit{large deviation} estimates for the process $a$ are available. Using our method, among others, we obtain the optimal rates of small deviations in supremum and $β$- Hölder norms for fractional Brownian motion with Hurst parameter $H\le\ \frac{1}{2}$. As an application, we discuss the usefulness of our upper bounds for small deviations in pathwise stochastic integral representation of random variables motivated by the hedging problem in mathematical finance.

math.PR↗

arXiv2014

The law of iterated logarithm for subordinated Gaussian sequences: uniform Wasserstein bounds

We develop a new method for showing that a given sequence of random variables verifies an appropriate law of the iterated logarithm. Our tools involve the use of general estimates on multidimensional Wasserstein distances, that are in turn based on recently developed inequalities involving Stein matrices and transport distances. Our main application consists in the proof of the exact law of the iterated logarithm for the Hermite variations of a fractional Brownian motion in the critical case.

math.PR↗

arXiv2014

Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators $Γ_i$ (introduced by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly \cite{n-po-1}, concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

math.PR↗

arXiv2014

Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind

Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -θX_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ θ>0$ with Gaussian driving noise $ Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}$, where $ a_t= H e^{\frac{t}{H}}$ and $B$ is a fractional Brownian motion with Hurst parameter $H \in (0,1)$. In this article, we consider the case $H>\frac{1}{2}$. Then using the ergodicity of $\text{fOU}_{2}$ process, we construct consistent estimators of drift parameter $θ$ based on discrete observations in two possible cases: $(i)$ the Hurst parameter $H$ is known and $(ii)$ the Hurst parameter $H$ is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range $H \in (\frac{1}{2},1)$.

math.PR↗

arXiv2014

Necessary and Sufficient Conditions for Hölder Continuity of Gaussian Processes

The continuity of Gaussian processes is extensively studied topic and it culminates in the Talagrand's notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov-Čentsov condition.

math.PR↗

arXiv2013

Fourth Moment Theorems for Markov Diffusion Generators

Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.

math.PR↗

arXiv2013

Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the Second Kind

Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -θX_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ θ>0$ with driving noise $ Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}; \ a_t= H e^{\frac{t}{H}}$ where $B$ is a fractional Brownian motion with Hurst parameter $H \in (0,1)$. In this article, in the case $H>1/2$, we prove that the least squares estimator $\hatθ_T$ introduced in [\cite{h-n}, Statist. Probab. Lett. 80, no. 11-12, 1030-1038], provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range $H \in(1/2,1)$.

math.PR↗

arXiv2012

Rate of convergence for discretization of integrals with respect to Fractional Brownian motion

In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$ is considered. In Statistics & Decisions, 27, 129-143, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as possible to $H - 1/2$.

math.PR↗

arXiv2012

Spectral characterization of the quadratic variation of mixed Brownian fractional Brownian motion

Dzhaparidze and Spreij [5] showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. [2], where it is shown that the quadratic variation of the log-returns determines the hedging strategy.

math.PR↗

arXiv2010

When does fractional Brownian motion not behave as a continuous function with bounded variation?

If we compose a smooth function g with fractional Brownian motion B with Hurst index H > 1/2, then the resulting change of variables formula [or It/^o- formula] has the same form as if fractional Brownian motion would be a continuous function with bounded variation. In this note we prove a new integral representation formula for the running maximum of a continuous function with bounded variation. Moreover we show that the analogue to fractional Brownian motion fails.

math.PR↗

arXiv2010

On the fractional Black-Scholes market with transaction costs

We consider fractional Black-Scholes market with proportional transaction costs. When transaction costs are present, one trades periodically i.e. we have the discrete trading with equidistance $n^{-1}$ between trading times. We derive a non trivial hedging error for a class of European options with convex payoff in the case when the transaction costs coefficients decrease as $n^{-(1-H)}$. We study the expected hedging error and asymptotic behavior of the hedge as $H \to 1/2$

q-fin.PR↗