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Bounds and large deviation results for boundary non-crossing probabilities of Gaussian processesMar 14 2019We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm{P}\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by arbitrary compact separable metrizable space $\mathbb T$. We obtain upper ... More

Betti Numbers of Gaussian Excursions in the Sparse RegimeJul 29 2018Aug 23 2018Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise ... More

Improvements on the distribution of maximal segmental scores in a Markovian sequenceMar 07 2018Let $(A_i)_{i \geq 0}$ be a finite state irreducible aperiodic Markov chain and $f$ a lattice score function such that the average score is negative and positive scores are possible. Define $S_0:=0$ and $S_k:=\sum_{i=1}^k f(A_i)$ the successive partial ... More

A strong averaging principle for Lévy diffusions in foliated spaces with unbounded leavesFeb 02 2018Apr 10 2018This article extends a strong averaging principle for L\'evy diffusions which live on the leaves of a foliated manifold subject to small transversal L\'evy type perturbation to the case of non-compact leaves. The main result states that the existence ... More

A strong averaging principle for Lévy diffusions in foliated spaces with unbounded leavesFeb 02 2018This article extends a strong averaging principle for L\'evy diffusions which live on the leaves of a foliated manifold subject to small transversal L\'evy type perturbation to the case of non-compact leaves. The main result states that the existence ... More

Wolf Barth (1942--2016)Apr 24 2017In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his ... More

A self-calibrating method for heavy tailed data modelling. Application in neuroscience and financeDec 12 2016Dec 26 2017Modelling non-homogeneous and multi-component data is a problem that challenges scientific researchers in several fields. In general, it is not possible to find a simple and closed form probabilistic model to describe such data. That is why one often ... More

On a consistent estimator of a useful signal in Ornstein-Uhlenbeck model in $\mathbb{C}[-l,l[$Dec 10 2016Dec 17 2016~It is considered a transmittion process of a useful signal in Ornstein-Uhlenbeck model in $\mathbb{C}[-l,l[$ defined by the stochastic differential equation $$ d\Psi(t,x,\omega)=\sum_{n=0}^{2m} A_n\frac{\partial^{n}}{\partial x^{n}}\Psi(t,x,\omega)dt ... More

Large excursions and conditioned laws for recursive sequences generated by random matricesAug 18 2016We determine the large exceedance probabilities and large exceedance paths for the matrix recursive sequence $V_n = M_n V_{n-1} + Q_n, \: n=1,2,\ldots,$ where $\{M_n\}$ is an i.i.d. sequence of $d \times d$ random matrices and $\{ Q_n\}$ is an i.i.d. ... More

Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic processAug 16 2016Aug 28 2016It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-\theta t} + \mu (1-e^{-\theta t}) + \sigma \int_0^t e^{-\theta (t-s)} dW_s$, where $x_0 \in R$, $\theta>0$, $ \mu \in R$ and $\sigma > 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding ... More

Risk contagion under regular variation and asymptotic tail independenceMar 30 2016Risk contagion concerns any entity dealing with large scale risks. Suppose (X,Y) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of ... More

Risk contagion under regular variation and asymptotic tail independenceMar 30 2016Apr 25 2017Risk contagion concerns any entity dealing with large scale risks. Suppose (X,Y) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of ... More

Climbing down Gaussian peaksJan 28 2015How likely is the high level of a continuous Gaussian random field on an Euclidean space to have a "hole" of a certain dimension and depth? Questions of this type are difficult, but in this paper we make progress on questions shedding new light in existence ... More

The Three-body problem and the shape sphereFeb 04 2014[This is an expository article. I have submitted it to the American Mathematical Monthly.] The three-body problem defines a dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar ... More

On the application of McDiarmid's inequality to complex systemsAug 15 2013McDiarmid's inequality has recently been proposed as a tool for setting margin requirements for complex systems. If $F$ is the bounded output of a complex system, depending on a vector of $n$ bounded inputs, this inequality provides a bound $B_F(\epsilon)$, ... More

Ptolemy diagrams and torsion pairs in the cluster categories of Dynkin type DMar 07 2013We give a complete classification of torsion pairs in the cluster category of Dynkin type D_n, via a bijection to new combinatorial objects called Ptolemy diagrams of type D. For the latter we give along the way different combinatorial descriptions. One ... More

Concentration inequalities for order statisticsJul 31 2012This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs to a maximum ... More

Rare event simulation for processes generated via stochastic fixed point equationsJul 17 2011Jul 03 2014In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable $V$ satisfying the distributional equation $V\stackrel{\mathcal{D}}{=}f(V)$, where $f(v)=A\max\{v,D\}+B$ ... More

A non-simply laced version for cluster structures on 2-Calabi-Yau categoriesOct 27 2009Dec 06 2013This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting subcategory with ... More

A Spectral Analysis of the Sequence of Firing Phases in Stochastic Integrate-and-Fire OscillatorsJul 21 2009Integrate and fire oscillators are widely used to model the generation of action potentials in neurons. In this paper, we discuss small noise asymptotic results for a class of stochastic integrate and fire oscillators (SIFs) in which the buildup of membrane ... More

Isotropic Ornstein-Uhlenbeck flowsNov 07 2008Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic flows which has been studied extensively by various authors. Their rich structure allows for explicit calculations in several situations and makes them a natural object to start ... More

Cutting Sequences and PalindromesMar 03 2008Jun 11 2008We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday.

Enumerating Palindromes and Primitives in Rank Two Free GroupsFeb 19 2008Feb 12 2011Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and backwards. It is ... More

Level crossings and other level functionals of stationary Gaussian processesDec 20 2006This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, ... More

Subexponential asymptotics of hybrid fluid and ruin modelsMar 23 2005We investigate the tail asymptotics of the supremum of X(t)+Y(t)-ct, where X={X(t),t\geq 0} and Y={Y(t),t\geq 0} are two independent stochastic processes. We assume that the process Y has subexponential characteristics and that the process X is more regular ... More