2,636 research outputs found
Maxima of branching random walks with piecewise constant variance
This article extends the results of Fang & Zeitouni (2012a) on branching
random walks (BRWs) with Gaussian increments in time inhomogeneous
environments. We treat the case where the variance of the increments changes a
finite number of times at different scales in [0,1] under a slight restriction.
We find the asymptotics of the maximum up to an OP(1) error and show how the
profile of the variance influences the leading order and the logarithmic
correction term. A more general result was independently obtained by Mallein
(2015b) when the law of the increments is not necessarily Gaussian. However,
the proof we present here generalizes the approach of Fang & Zeitouni (2012a)
instead of using the spinal decomposition of the BRW. As such, the proof is
easier to understand and more robust in the presence of an approximate
branching structure.Comment: 28 pages, 4 figure
Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex
Let and let , be
such that and . We prove that \begin{equation*} a \mapsto \frac{\Gamma(aM +
1)}{\prod_{i=1}^{d+1} \Gamma(a \gamma_i + 1)} \prod_{i=1}^{d+1} x_i^{a\gamma_i}
\end{equation*} is completely monotonic on . This result
generalizes the one found by Alzer (2018) for binomial probabilities ().
As a consequence of the log-convexity, we obtain some combinatorial
inequalities for multinomial coefficients. We also show how the main result can
be used to derive asymptotic formulas for quantities of interest in the context
of statistical density estimation based on Bernstein polynomials on the
-dimensional simplex.Comment: 7 pages, 0 figur
Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance
We continue our study of the scale-inhomogeneous Gaussian free field
introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free
energy on V_N and adapt a technique of Bovier and Kurkova (2004b) to determine
the limiting two-overlap distribution. The adaptation was already successfully
applied in the simpler case of Arguin and Zindy (2015), where the limiting free
energy was computed for the field with two levels (in the center of V_N) and
the limiting two-overlap distribution was determined in the homogeneous case.
Our results agree with the analogous quantities for the Generalized Random
Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova
(2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra
identities hold exactly in the limit. As a corollary, the limiting array of
overlaps is ultrametric and the limiting Gibbs measure has the same law as a
Ruelle probability cascade.Comment: 52 pages, 6 figure
Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function
In Arguin & Tai (2018), the authors prove the convergence of the two-overlap
distribution at low temperature for a randomized Riemann zeta function on the
critical line. We extend their results to prove the Ghirlanda-Guerra
identities. As a consequence, we find the joint law of the overlaps under the
limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is
expected that we can adapt the approach to prove the same result for the
Riemann zeta function itself.Comment: 15 pages, 1 figur
Large deviations and continuity estimates for the derivative of a random model of on the critical line
In this paper, we study the random field \begin{equation*} X(h) \circeq
\sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1],
\end{equation*} where is an i.i.d. sequence of
uniform random variables on the unit circle in . Harper (2013)
showed that is a good model for the large values of
when is large, if
we assume the Riemann hypothesis. The asymptotics of the maximum were found in
Arguin, Belius & Harper (2017) up to the second order, but the tightness of the
recentered maximum is still an open problem. As a first step, we provide large
deviation estimates and continuity estimates for the field's derivative
. The main result shows that, with probability arbitrarily close to ,
\begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1),
\end{equation*} where a discrete set containing points.Comment: 7 pages, 0 figur
Extremes of the two-dimensional Gaussian free field with scale-dependent variance
In this paper, we study a random field constructed from the two-dimensional
Gaussian free field (GFF) by modifying the variance along the scales in the
neighborhood of each point. The construction can be seen as a local martingale
transform and is akin to the time-inhomogeneous branching random walk. In the
case where the variance takes finitely many values, we compute the first order
of the maximum and the log-number of high points. These quantities were
obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when
the variance is constant on all scales. The proof relies on a truncated second
moment method proposed by Kistler (2015), which streamlines the proof of the
previous results. We also discuss possible extensions of the construction to
the continuous GFF.Comment: 30 pages, 4 figures. The argument in Lemma 3.1 and 3.4 was revised.
Lemma A.4, A.5 and A.6 were added for this reason. Other typos were corrected
throughout the article. The proof of Lemma A.1 and A.3 was simplifie
A uniform law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator
We develop a new law of large numbers where the -th summand is given
by a function evaluated at , and where is an estimator converging in probability
to some parameter . Under broad technical conditions, the
convergence is shown to hold uniformly in the set of estimators interpolating
between and another consistent estimator . Our main
contribution is the treatment of the case where blows up at , which is
not covered by standard uniform laws of large numbers.Comment: 10 pages, 1 figur
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