Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of centra limit theorems with infinite variance stable limits. We illustrate the principle for \sv\ models, functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations

The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains

We introduce the cluster index of a multivariate regularly varying stationary sequence and characterize the index in terms of the spectral tail process. This index plays a major role in limit theory for partial sums of regularly varying sequences. We illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary Markov chain under a drift condition

Entropy and the Law of Small Numbers

Two new information-theoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary information-theoretic techniques it is shown that, when $S_n=\sum_{i=1}^nX_i$ is the sum of the (possibly dependent) binary random variables $X_1,X_2,...,X_n$, with $E(X_i)=p_i$ and E(S_n)=\la, then \ben D(P_{S_n}\|\Pol)\leq \sum_{i=1}^n p_i^2 + \Big[\sum_{i=1}^nH(X_i) - H(X_1,X_2,..., X_n)\Big], \een where D(P_{S_n}\|{Po}(\la)) is the relative entropy between the distribution of $S_n$ and the Poisson(\la) distribution. The first term in this bound measures the individual smallness of the $X_i$ and the second term measures their dependence. A general method is outlined for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution. Second, in the particular case when the $X_i$ are independent, the following sharper bound is established, \ben D(P_{S_n}\|\Pol)\leq \frac{1}{\lambda} \sum_{i=1}^n \frac{p_i^3}{1-p_i}, % \label{eq:abs2} \een and it is also generalized to the case when the $X_i$ are general integer-valued random variables. Its proof is based on the derivation of a subadditivity property for a new discrete version of the Fisher information, and uses a recent logarithmic Sobolev inequality for the Poisson distribution.Comment: 15 pages. To appear, IEEE Trans Inform Theor

H\"older-type inequalities and their applications to concentration and correlation bounds

Let $Y_v, v\in V,$ be $[0,1]$-valued random variables having a dependency graph $G=(V,E)$. We show that $$\mathbb{E}\left[\prod_{v\in V} Y_{v} \right] \leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{\chi_b}{b}}\right] \right\}^{\frac{b}{\chi_b}},$$ where $\chi_b$ is the $b$-fold chromatic number of $G$. This inequality may be seen as a dependency-graph analogue of a generalised H\"older inequality, due to Helmut Finner. Additionally, we provide applications of H\"older-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables.Comment: 15 page

Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke

Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p_n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavytailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.Comment: in Extremes; Statistical Theory and Applications in Science, Engineering and Economics; ISSN 1386-1999; (2016