Related papers: Large Deviations Application to Billingsley's Exam…

We extend the Matom\"{a}ki-Radziwi\l\l{} theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a…

The standard Large Deviation Theory (LDT) mirrors the Boltzmann-Gibbs (BG) factor which describes the thermal equilibrium of short-range Hamiltonian systems, the velocity distribution of which is Maxwellian. It is generically applicable to…

In this paper we prove a Large Deviation Principle for the sequence of symmetrised empirical measures $\frac{1}{n} \sum_{i=1}^{n} \delta_{(X^n_i,X^n_{\sigma_n(i)})}$ where $\sigma_n$ is a random permutation and $((X_i^n)_{1 \leq i \leq…

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly…

We consider the Fleming--Viot particle system associated with a continuous-time Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its…

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the…

In this paper we propose the use of $\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process $\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t$, from discrete…

For a regression model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and $\chi^2$ divergence. The…

We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_1(n)),\ldots, X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced…

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not…

We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function L_n(z)=|W_n|^{-1}\logE\exp{z|\Xi\cap W_n|} of the empirical volume fraction |\Xi\cap W_n|/|W_n|, where |\cdot| denotes the d-dimensional Lebesgue…

A sharp, distribution free, non-asymptotic result is proved for the concentration of a random function around the mean function, when the randomization is generated by a finite sequence of independent data and the random functions satisfy…

$f$-divergences, which quantify discrepancy between probability distributions, are ubiquitous in information theory, machine learning, and statistics. While there are numerous methods for estimating $f$-divergences from data, a limit…

Suppose $\widehat\theta_n$ is a strongly consistent estimator for $\theta_0$ in some i.i.d. situation. Let $N_\varepsilon$ and $Q_\varepsilon$ be respectively the last $n$ and the total number of $n$ for which $\widehat\theta_n$ is at least…

Consider $n$ iid random variables, where $\xi_1, \ldots, \xi_n$ are $n$ realisations of a random variable $\xi$ and $\zeta_1, \ldots, \zeta_n$ are $n$ realisations of a random variable $\zeta$. The distribution of each realisation of $\xi$,…

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization…

We prove a deviation bound for the maximum of partial sums of functions of $\alpha$-dependent sequences as defined in Dedecker, Gou{\"e}zel and Merlev{\`e}de (2010). As a consequence, we extend the Rosenthal inequality of Rio (2000) for…

We use Stein's method to prove a generalization of the Lindeberg-Feller CLT providing an upper and a lower bound for the superior limit of the Kolmogorov distance between a normally distributed random variable and the rowwise sums of a…

The Airy$_\beta$ point process, $a_i \equiv N^{2/3} (\lambda_i-2)$, describes the eigenvalues $\lambda_i$ at the edge of the Gaussian $\beta$ ensembles of random matrices for large matrix size $N \to \infty$. We study the probability…

This paper considers estimation of large dynamic factor models with common and idiosyncratic trends by means of the Expectation Maximization algorithm, implemented jointly with the Kalman smoother. We show that, as the cross-sectional…