28 research outputs found
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . 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
Weighted dependency graphs
The theory of dependency graphs is a powerful toolbox to prove asymptotic
normality of sums of random variables. In this article, we introduce a more
general notion of weighted dependency graphs and give normality criteria in
this context. We also provide generic tools to prove that some weighted graph
is a weighted dependency graph for a given family of random variables.
To illustrate the power of the theory, we give applications to the following
objects: uniform random pair partitions, the random graph model ,
uniform random permutations, the symmetric simple exclusion process and
multilinear statistics on Markov chains. The application to random permutations
gives a bivariate extension of a functional central limit theorem of Janson and
Barbour. On Markov chains, we answer positively an open question of Bourdon and
Vall\'ee on the asymptotic normality of subword counts in random texts
generated by a Markovian source.Comment: 57 pages. Third version: minor modifications, after review proces
The number of flags in finite vector spaces: Asymptotic normality and Mahonian statistics
We study the generalized Galois numbers which count flags of length r in
N-dimensional vector spaces over finite fields. We prove that the coefficients
of those polynomials are asymptotically Gaussian normally distributed as N
becomes large. Furthermore, we interpret the generalized Galois numbers as
weighted inversion statistics on the descent classes of the symmetric group on
N elements and identify their asymptotic limit as the Mahonian inversion
statistic when r approaches infinity. Finally, we apply our statements to
derive further statistical aspects of generalized Rogers-Szegoe polynomials,
re-interpret the asymptotic behavior of linear q-ary codes and characters of
the symmetric group acting on subspaces over finite fields, and discuss
implications for affine Demazure modules and joint probability generating
functions of descent-inversion statistics.Comment: 19 pages. Corrected proof of asymptotic normality (Theorem 3.5).
Previous Proposition 3.3 is fals
Symbolic Calculus in Mathematical Statistics: A Review
In the last ten years, the employment of symbolic methods has substantially
extended both the theory and the applications of statistics and probability.
This survey reviews the development of a symbolic technique arising from
classical umbral calculus, as introduced by Rota and Taylor in The
usefulness of this symbolic technique is twofold. The first is to show how new
algebraic identities drive in discovering insights among topics apparently very
far from each other and related to probability and statistics. One of the main
tools is a formal generalization of the convolution of identical probability
distributions, which allows us to employ compound Poisson random variables in
various topics that are only somewhat interrelated. Having got a different and
deeper viewpoint, the second goal is to show how to set up algorithmic
processes performing efficiently algebraic calculations. In particular, the
challenge of finding these symbolic procedures should lead to a new method, and
it poses new problems involving both computational and conceptual issues.
Evidence of efficiency in applying this symbolic method will be shown within
statistical inference, parameter estimation, L\'evy processes, and, more
generally, problems involving multivariate functions. The symbolic
representation of Sheffer polynomial sequences allows us to carry out a
unifying theory of classical, Boolean and free cumulants. Recent connections
within random matrices have extended the applications of the symbolic method.Comment: 72 page