Related papers: On the Central Limit Theorem for the log-partition…

We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of…

Let $p_n(y)=\sum_k\hat{\alpha}_k\phi(y-k)+\sum_{l=0}^{j_n-1}\sum_k\hat {\beta}_{lk}2^{l/2}\psi(2^ly-k)$ be the linear wavelet density estimator, where $\phi$, $\psi$ are a father and a mother wavelet (with compact support),…

We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled…

For a Brownian directed polymer in a Gaussian random environment, with $q(t,\cdot)$ denoting the quenched endpoint density and \[ Q_n(t,x_1,\ldots,x_n)=\mathbf{E}[q(t,x_1)\ldots q(t,x_n)], \] we derive a hierarchical PDE system satisfied by…

We present results about large deviations and laws of large numbers for various polymer related quantities. In a completely general setting and strictly positive temperature, we present results about large deviations for directed polymers…

We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of…

Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing with the sample size have received a lot of attention in the recent times. Chernozhukov et al. (2017) proved a Berry--Esseen type result for…

We prove a conjecture of Toth and Veto about the weak convergence of the self repelling random walk with directed edges under diffusive scaling to a uniform distribution.

The stratified resampling mechanism is one of the resampling schemes commonly used in the resampling steps of particle filters. In the present paper, we prove a central limit theorem for this mechanism under the assumption that the initial…

The applicability conditions of a recently reported Central Limit Theorem-based approximation method in statistical physics are investigated and rigorously determined. The failure of this method at low and intermediate temperature is proved…

We study the adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erd\H{o}s-R\'enyi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the…

In this work, we obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the "spectrum" of partitions of a large integer n (under the Plancherel measure). More specifically, we show that,…

We consider a one-dimensional directed polymer in a random potential which is characterized by the Gaussian statistics with the finite size local correlations. It is shown that the well-known Kardar's solution obtained originally for a…

In these proceedings, we first summarize some general properties of phase transitions in the presence of quenched disorder, with emphasis on the following points: the need to distinguish typical and averaged correlations, the possible…

We prove the Central Limit Theorem and superpolynomial mixing for environment viewed for the particle process in quasi periodic Diophantine random environment. The main ingredients are smoothness estimates for the solution of the Poisson…

We study the individual digits for the absolute value of the characteristic polynomial for the Circular $\beta$-Ensemble. We show that, in the large $N$ limit, the first digits obey Benford's Law and the further digits become uniformly…

In this work, we consider a wide two-layer neural network and study the behavior of its empirical weights under a dynamics set by a stochastic gradient descent along the quadratic loss with mini-batches and noise. Our goal is to prove a…

We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $\mu^{(r)}(d)$ as the density of integers $n\in…

Given a graph $G = (V,E)$, an $(\alpha, \beta)$-ruling set is a subset $S \subseteq V$ such that the distance between any two vertices in $S$ is at least $\alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is…

In this paper we study asymptotic properties of symmetric and non-degenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous…