We consider a one-dimensional, nearest-neighbour, recurrent random walk (Xk), k <= n, in a random i.i.d. environment (RWRE). Assuming that the environment has finite support, which is treated as a parameter, we establish the Local Asymptotic Mixed Normality (LAMN) property for this parameter. The convergence rate is root n, and the asymptotic Fisher information is random and expressed in terms of the invariant measure in the infinite valley, as introduced in [N. Gantert, Y. Peres, and Z. Shi, Ann. Inst. Henri Poincar & eacute; Probab. Stat. 46 (2010) 525-536]. We further show that the Maximum Likelihood Estimator (MLE) of the support parameter converges to a mixture of normal distributions and is asymptotically efficient.
The Weighted Stochastic Block Model is a statistical model for unsupervised clustering of individuals based on a dissimilarity matrix. The computation of the binary conditional marginals of group memberships by exact enumeration is out of reach. We use the fact that the joint conditional distribution, given the dissimilarities, is factored into binary factors and we describe how to obtain a Tensor-Train (TT) decomposition from the SVD of each factor. It leads to variable separation involving products of matrices and we show how to exploit it for the computation of binary conditional marginals. This exact calculation faces severe numerical challenges. The dimensions of the matrices involved in the products increase exponentially with the number of factors. We bypass this difficulty by using the TT-matrix format. The TT-rank of the products themselves grows exponentially. We use a numerical approximation called rounding by Oseledets to control this. We compare the behavior of the TT approach with the Mean-Field approximation and the Gibbs Sampler for various dissimilarity structures and up to forty variables. We recommend the TT approach due to its accuracy and reasonable computing time.
In order to guarantee the reliability of industrial systems, statistical studies can be carried out on a variable of interest, Y, or, more precisely, on a risk measure computed based on this variable. This article studies and compares different risk measures, on the one hand, according to their theoretical properties, and, on the other hand, according to their numerical estimation. In particular, a distinction is proposed between risk measures homogeneous to the variable of interest Y (margin, quantile, and superquantile) and risk measures homogeneous to a probability (failure and buffered failure probability). More specifically, a duality between these two categories is developed, showing the equivalence of using a Y-homogeneous risk measure with a dual, homogeneous, [0,1]-homogeneous risk measure. Finally, a non-asymptotic statistical framework is proposed for decision-making using these risk measures. The latter can be used in practice in the case where the considered risk measures are the probability of failure and the quantile, and is equivalent, under one assumption, to a quantile estimation using the Wilks' method.
We consider a one-dimensional, nearest-neighbour, recurrent random walk $(X_k)$, $k\leq n$, in a random i.i.d. environment (RWRE). Assuming that the environment has finite support, which is treated as a parameter, we establish the Local Asymptotic Mixed Normality (LAMN) property for this parameter. The convergence rate is $\sqrt n$, and the asymptotic Fisher information is random and expressed in terms of the invariant measure in the infinite valley, as introduced in Gantert et al., 2010. We further show that the Maximum Likelihood Estimator (MLE) of the support parameter converges to a mixture of normal distributions and is asymptotically efficient. We further show that the Maximum Likelihood Estimator (MLE) of the support parameter converges to a mixture of normal distributions and is asymptotically efficient. The proofs are based on a recent result by Comets et al.,2024, which extends to recurrent RWRE the method of the ``environment viewed from the particle'', originally introduced in \cite{KozMol} for the transient ballistic RWRE.
Calibration of sensors in partially observed and correlated environments raises fundamental challenges for variable selection and model interpretation. When observations are noisy and influenced by unmeasured factors, classical selection criteria based on regression coefficients, crossvalidation errors, or sensitivity indices may fail to identify the variables that most effectively reduce uncertainty in the target quantity. This paper introduces a probabilistic framework for variable selection based on variance reduction and prediction stability. The proposed approach relies on the systematic evaluation of models built from different subsets of observed variables and on a criterion that minimizes conditional prediction variance under a parsimony constraint. This criterion naturally penalizes spurious correlations and distinguishes variables that contribute to uncertainty reduction from those that merely compensate for missing information. The framework is illustrated through analytical examples and numerical experiments on simulated data, which highlight its robustness to noise and unmeasured confounders. Although motivated by calibration problems in environmental sensing, the proposed methodology is general and applicable to a wide range of regression and inference tasks involving correlated inputs and unobserved factors.
