arXiv subjects

Sana Louhichi

Publications and source records attributed to Sana Louhichi.

16 recordsLinked to original sources

arXiv2026

Gumbel convergence for maximal packing distances from uniform random samples

Let $(X _n )_n$ be a sequence of i.i.d. $R^d$-valued random variables uniformly distributed on a compact subset $M$ of $R^d$ . In this work, we study the asymptotic behavior of maximal packings of this compact subset $M$. Under some assumptions on $M$ (in particular, that it is a smooth submanifold of $R^d$ without boundary), we establish a precise extreme value law (of Gumbel type) for the minimal distances from the sample $(X_1 , \cdots, X_ n)$ to the centers of maximal packings. This result leads to explicit asymptotic confidence bounds for this maximal packing and thus to the support $M$. A distinctive feature of our contribution is the explicit derivation of the scaling sequences in the Gumbel convergence, depending only on the geometry of the support M. These formulas provide interpretable confidence bounds for $M$, which represent a novel complement to previous approaches such as those by Fasy et al. (2014). Our approach bridges geometric and combinatorial probability arguments and relies on analytic tools such as the Lambert $W$ function. In addition, we provide examples of main sample spaces in directional and circular statistics (circle, sphere, torus), along with simulations that illustrate and support the theoretical findings.

math.PR↗

arXiv2023

Topological reconstruction of compact supports of dependent stationary random variables

In this paper we extend results on reconstruction of probabilistic supports of random i.i.d variables to supports of dependent stationary $\mathbb R^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.

math.PR↗

arXiv2022

Self-Training of Halfspaces with Generalization Guarantees under Massart Mislabeling Noise Model

We investigate the generalization properties of a self-training algorithm with halfspaces. The approach learns a list of halfspaces iteratively from labeled and unlabeled training data, in which each iteration consists of two steps: exploration and pruning. In the exploration phase, the halfspace is found sequentially by maximizing the unsigned-margin among unlabeled examples and then assigning pseudo-labels to those that have a distance higher than the current threshold. The pseudo-labeled examples are then added to the training set, and a new classifier is learned. This process is repeated until no more unlabeled examples remain for pseudo-labeling. In the pruning phase, pseudo-labeled samples that have a distance to the last halfspace greater than the associated unsigned-margin are then discarded. We prove that the misclassification error of the resulting sequence of classifiers is bounded and show that the resulting semi-supervised approach never degrades performance compared to the classifier learned using only the initial labeled training set. Experiments carried out on a variety of benchmarks demonstrate the efficiency of the proposed approach compared to state-of-the-art methods.

cs.LG↗

arXiv2021

On bandwidth selection problems in nonparametric trend estimation under martingale difference errors

In this paper, we are interested in the problem of smoothing parameter selection in nonparametric curve estimation under dependent errors. We focus on kernel estimation and the case when the errors form a general stationary sequence of martingale difference random variables where neither linearity assumption nor "all moments are finite" are required.We compare the behaviors of the smoothing bandwidths obtained by minimizing either the unknown average squared error, the theoretical mean average squared error, a Mallows-type criterion adapted to the dependent case and the family of criteria known as generalized cross validation (GCV) extensions of the Mallows' criterion. We prove that these three minimizers and those based on the GCV family are first-order equivalent in probability. We give also a normal asymptotic behavior of the gap between the minimizer of the average square error and that of the Mallows-type criterion. This is extended to the GCV family.Finally, we apply our theoretical results to a specific case of martingale difference sequence, namely the Auto-Regressive Conditional Heteroscedastic (ARCH(1)) process.A Monte-carlo simulation study, for this regression model with ARCH(1) process, is conducted.

math.ST↗

arXiv2019

The reproducing kernel Hilbert space approach in nonparametric regression problems with correlated observations

In this paper we investigate the problem of estimating the regression function in models with correlated observations. The data is obtained from several experimental units each of them forms a time series. We propose a new estimator based on the inverse of the autocovariance matrix of the observations, assumed known and invertible. Using the properties of the Reproducing Kernel Hilbert spaces, we give the asymptotic expressions of its bias and its variance. In addition, we give a theoretical comparison, by calculating the IMSE, between this new estimator and the classical one proposed by Gasser and Muller. Finally, we conduct a simulation study to investigate the performance of the proposed estimator and to compare it to the Gasser and Muller's estimator in a finite sample set.

math.ST↗

arXiv2018

Dynamical coupling between Ising and FK percolation

We investigate the problem of constructing a dynamics on edge--spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations. We dream of constructing a Markov process on edge--spin configurations which is reversible with respect to the Ising--FK coupling measure, and such that the marginal on the spins is a Glauber dynamics, while the marginal on the edges is a Markovian evolution. We present two local dynamics, one which fulfills only the first condition and one which fulfills the first two conditions. We show next that our dream process is not feasible in general. We present a third dynamics, which is non local and fulfills the first and the third conditions. We finally present a localized version of this third dynamics, which can be seen as a contraction of the first dynamics.

math.PR↗

arXiv2017

Multiplicative ergodicity of laplace transforms for additive functional of markov chains with application to age-dependent branching process

We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).

math.PR↗

arXiv2014

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients necessary and sufficient conditions for the finite dimensional convergence to an $α$-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $α\leq 1$. In the functional setting we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $S$ topology, introduced by Jakubowski (1997).

math.PR↗

arXiv2013

Asymptotic equivalence of jumps Lévy processes and their discrete counterpart

We establish the global asymptotic equivalence between a pure jumps Lévy process $\{X_t\}$ on the time interval $[0,T]$ with unknown Lévy measure $ν$ belonging to a non-parametric class and the observation of $2m^2$ Poisson independent random variables with parameters linked with the Lévy measure $ν$. The equivalence result is asymptotic as $m$ tends to infinity. The time $T$ is kept fixed and the sample path is continuously observed. This result justifies the idea that, from a statistical point of view, knowing how many jumps fall into a grid of intervals gives asymptotically the same amount of information as observing $\{X_t\}$.

math.PR↗

arXiv2013

Exponential growth of bifurcating processes with ancestral dependence

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the i.i.d. supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

q-bio.PE↗

arXiv2012

Functional Convergence of Linear Sequences in a non-Skorokhod Topology

In this article, we prove a new functional limit theorem for the partial sum sequence $S_{[nt]}=\sum_{i=1}^{[nt]}X_i$ corresponding to a linear sequence of the form $X_i=\sum_{j \in \bZ}c_j ξ_{i-j}$ with i.i.d. innovations $(ξ_i)_{i \in \bZ}$ and real-valued coefficients $(c_j)_{j \in \bZ}$. This weak convergence result is obtained in space $\bD[0,1]$ endowed with the $S$-topology introduced in Jakubowski (1992), and the limit process is a linear fractional stable motion (LFSM). One of our result provides an extension of the results of Avram and Taqqu (1992) to the case when the coefficients $(c_j)_{j \in \bZ}$ may not have the same sign. The proof of our result relies on the recent criteria for convergence in Skorokhod's $M_1$-topology (due to Louhichi and Rio (2011)), and a result which connects the weak $S$-convergence of the sum of two processes with the weak $M_1$-convergence of the two individual processes. Finally, we illustrate our results using some examples and computer simulations.

math.PR↗

arXiv2010

A Cluster Limit Theorem for Infinitely Divisible Point Processes

In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in distribution of $(N_n)_{n \geq 1}$ to an infinitely divisible point process $N$. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.

math.PR↗

arXiv2009

Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays

In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n, n \geq 1}$ of random variables with values in $\bR \verb2\2 \{0\}$ (which satisfy some asymptotic dependence conditions), and the corresponding sequence $(N_{n})_{n\geq 1}$ of point processes, where $N_{n}$ has the points $X_{j,n}, 1\leq j \leq n$. Our main result identifies some explicit conditions for the convergence of the sequence $(N_{n})_{n \geq 1}$, in terms of the probabilistic behavior of the variables in the array.

math.PR↗

arXiv2008

Convergence of Point Processes with Weakly Dependent Points

For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_n=\sum_{j=1}^{n}δ_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence, we obtain the convergence in distribution of the partial sum sequence $S_n=\sum_{j=1}^{n}X_{j,n}$ to an infinitely divisible random variable, whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.

math.PR↗

arXiv2007

A functional central limit theorem for interacting particle systems on transitive graphs

A finite range interacting particle system on a transitive graph is considered. Assuming that the dynamics and the initial measure are invariant, the normalized empirical distribution process converges in distribution to a centered diffusion process. As an application, a central limit theorem for certain hitting times, interpreted as failure times of a coherent system in reliability, is derived.

math-ph↗

arXiv2005

The initial drift of a 2D droplet at zero temperature

We consider the 2D stochastic Ising model evolving according to the Glauber dynamics at zero temperature. We compute the initial drift for droplets which are suitable approximations of smooth domains. A specific spatial average of the derivative at time~0 of the volume variation of a droplet close to a boundary point is equal to its curvature multiplied by a direction dependent coefficient. We compute the explicit value of this coefficient.

math.PR↗