Related papers: Large Deviation Principles for Lacunary Sums

We study the large deviation behavior of lacunary sums $(S_n/n)_{n\in \mathbb{N} }$ with $S_n:= \sum_{k=1}^n f(a_kU)$, $n\in\mathbb{N}$, where $U$ is uniformly distributed on $[0,1]$, $(a_k)_{k\in\mathbb{N}}$ is an Hadamard gap sequence,…

In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $\sum_{n=1}^N \cos(2 \pi n_k x)$, where the sequence $(n_k)_{k \geq 1}$ satisfies the Hadamard…

We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For $N \in \mathbb N$, we consider random $N \times N$ symmetric matrices $H^N$ which are such that…

A classical observation in analysis asserts that lacunary systems of dilated functions show many properties which are also typical for systems of independent random variables. For example, if $(n_k)_{k \ge 1}$ is a sequence of integers…

In 1975 Walter Philipp proved the law of the iterated logarithm (LIL) for the discrepancy of lacunary sequences: for any sequence $(n_k)_{k \geq 1}$ satisfying the Hadamard gap condition $n_{k+1} / n_k \geq q > 1,~k \geq 1,$ we have $$…

We consider the moment space $\mathcal{M}_n^{K}$ corresponding to $p \times p$ complex matrix measures defined on $K$ ($K=[0,1]$ or $K=\D$). We endow this set with the uniform law. We are mainly interested in large deviations principles…

We consider the standard first passage percolation model on $\mathbb Z^d$ with bounded and bounded away from zero weights. We show that the rescaled passage time $\widetilde{\mathbf T}_{n,X}$ restricted to a compact set $X$ satisfies a…

The d'Arcais polynomials $P_n(z)$ for $n\in\{0,1,\dots\}$ are defined as $\sum_{n=0}^{\infty} P_n(z) q^n = \exp(-z\ln((q;q)_{\infty}))$ where the $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k)$ for $|q|<1$. Denoting…

It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 \pi n_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k…

We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the…

Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial sum of Y_n just before that of X_n exceeds x>0. Motivated by stochastic models for neural activity, uniform convergence of the form $\sup_{c\in I}|a(c,x)\operatorname…

We establish large deviation principle (LDP) for the family of vector-valued random processes $(X^\epsilon,Y^\epsilon),\epsilon\to 0$ defined as $$ X^\epsilon_t=\frac{1}{\epsilon^\kappa}\int_0^t H(\xi^\epsilon_s,Y^\epsilon_s)ds,…

We prove pathwise large deviation principles of slow variables in slow-fast systems in the limit of time-scale separation tending to infinity. In the limit regime we consider, the convergence of the slow variable to its deterministic limit…

We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness…

Lacunary trigonometric and Walsh series satisfy limiting results that are typical for i.i.d. random variables such as the central limit theorem (Salem, Zygmund 1947), the law of the iterated logarithm (Weiss 1959) and several probability…

One says that the local large deviation principle (LLDP) is satisfied for a family of random vectors $\{\zeta_T\}_{T\ge 0}$ in $\mathbb R^d,$ $d\ge 1,$ if there exists a function $D:\mathbb R^d\to [0,\infty],$ $D\not \equiv \infty,$ such…

We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r(\tfrac{i}{N},\tfrac{j}{N})$,…

Denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to the classical $\ell_ p^n$-ball, defined as the set of…

Given lacunary sequence of integers, $n_k$, $n_{k+1}/n_k>\lambda>1$, we define a new sequence $\{m_k\}$ formed by all possible $l$-wise sums $\pm n_{k_1}\pm n_{k_2}\pm \ldots\pm n_{k_l}$. We prove if $\lambda>\lambda_l$, then any series…

We investigate possible large deviation principles (LDPs) for the $n$-vertex sampling from a given graphon with various speeds $s(n)$ and resolve all the cases except when the speed $s(n)$ is of order $n^2$. For quadratic speed…