Conference in Analysis and Probability
27 - 29 November 2015
Celebrating Professor B V Rao's 70th Birthday

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Lecture Abstracts

Antar Bandyopadhyay

De-Preferential Attachment Random Graphs

In this talk we will introduce a new model of a growing sequence of random graphs where a new vertex is less likely to join to an existing vertex with high degree and more likely to join to a vertex with low degree. In contrast to the well studied model of preferential attachment random graphs where higher degree vertices are preferred, we will call our model de-preferential attachment random graph model. We will consider two types of de-preferential attachment models, namely, inverse de-preferential, where the attachment probabilities are inversely proportional to the degree and linear de-preferential, where the attachment probabilities are proportional to \(c-\) degree, where \(c > 0\) is a constant. We will give asymptotic degree distribution for both the model and show that the limiting degree distribution has very thin tail. We will also show that for a fixed vertex \(v\), the degree grows as \(\sqrt{\log n}\) for the inverse de-preferential case and as \(\log n\) for the linear case, for a graph with \(n\) vertices. Some of the results will also be generalized when each new vertex joins to \(m > 1\) existing vertices.

[This is a joint work with Subhabrata Sen, Stanford University].

Arijit Chakrabarty

Minima of smooth Gaussian processes

In this work, we study the probability that the minimum of a Gaussian process, whose paths are sufficiently smooth, over a compact interval is larger than a high threshold. Adler et al. (2014) have studied this from the point of view of large deviations, and have obtained the decay rate on the logarithmic scale. We improve their result, under some additional assumptions, by obtaining the precise asymptotic rate. Furthermore, we study the asymptotic behaviour of the process conditioned on the event that the minimum is larger than a threshold, as the threshold goes to infinity. This is an ongoing joint work with Gennady Samorodnitsky.

Reference: R. J. Adler, E. Moldavskaya, and G. Samorodnitsky. On the existence of paths between points in high level excursion sets of Gaussian random fields. The Annals of Probability, 42(3):1020--1053, 2014.

Arup Bose

Free probability and high dimensional time series

Large dimensional autocovariance matrices arise naturally in the statistical analysis of high dimensional time series. We study the joint asymptotic behaviour of these matrices in an appropriate sense by establishing connection with free probability. We show how these results can be applied to inference problems such as white noise testing.
More generally, we show the convergence of elements of certain non-commutative space of matrices of increasing dimension generated by appropriate polynomials in non-random matrices and independent random matrices. These are interesting models in free probability in their own right. The resulting limits are described in terms of free circular, free semi-circular and other free variables.

[This is joint work with Monika Bhattacharjee]

Arup Pal

On the structure of certain \(C^*\)-algebra-representations

We will take a close look at the natural representation of the \(C^*\)-algebra corresponding to the quantum \(SU(n)\) group on its \(L_2\) space.

B Rajeev

Martingale Chaoses

Wiener's notion of `homogenuous chaos' for Brownian motion is one of the principal tools of stochastic analysis. It has been extended only to very special class of processes like the isonormal Gaussian processes, the normal martingales and Levy processes. We extend it to the class of continuous martingales, whose quadratic variations increase to infinity, by defining multiple stochastic integrals with respect to these martingales and discuss some consequences.

B V Rajarama Bhat

Random sets and tensor product systems of Hilbert spaces.

Tensor product systems of Hilbert spaces are useful in classifying semigroups of endomorphisms of the algebra of all bounded operators on a Hilbert space. B. Tsirelson obtained several different examples of such systems using stochastic processes or random sets such as zeros of a Brownian Motion. In the converse direction V. Liebscher obtained interesting random sets from pairs of product systems. We explain some of the basic ideas involved in these constructions.

K B Sinha

Brownian Bridge in Quantum Probability

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian Bridge implements the unitary isomorphism, the quantum Ito integrals wrt the creation and annihilation bridge processes give different left- and right- integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.

Koushik Saha

Fluctuations of linear eigenvalue statistics of band matrices

In this talk, we shall discuss the fluctuation of linear eigenvalue statistics of Random Band Matrices defined by \(M_n=\frac{1}{\sqrt {b_n}}W_n\), where \(W_n\) is a \(n\times n\) band Hermitian random matrix of bandwidth \(b_n\) and \(b_n\to\infty\) as \(n\to\infty\).

Krishanu Maulik

Urn models with random reinforcement weights

We consider an urn model where at each stage the replacement matrix is a random multiple of a fixed balanced non-negative one. The multipliers form an i.i.d. sequence of positive random variables. We discuss limit and mode of convergence of the colour proportions under various moment conditions on the multipliers and for different classes of the underlying balanced non-negative matrix. This is a joint work with Ujan Gangopadhyay.

Manjunath Krishnapur

Rigidity and tolerance of point processes

S. Ghosh and Y. Peres showed that in certain point processes, the configuration outside a bounded domain determines the number of points of the process inside the domain (and in some cases even the center of mass). They named this surprising phenomenon as rigidity.
In joint work with Subhroshekhar Ghosh, we systematically investigated the extent of rigidity and tolerance in determinantal point processes and in zeros of random analytic functions. I shall present some of the results and equally importantly, the definitions.

Parthanil Roy

Effective Dimension, Maximal Moments and Path Properties of Stable Fields

We give sharp upper bound on the rate of growth of maximal moments for a reasonably big class of stationary symmetric stable random fields. We shall also discuss the relationship between this rate of growth, effective dimension, and the uniform modulus of continuity of self-similar stable random fields with stationary increments. (This talk is based on a joint work with Snigdha Panigrahi and Yimin Xiao.)

Probal Chaudhuri

Statistical tests for high dimensional data with low sample sizes

For statistical tests involving data with large dimensions and relatively small sample sizes, the performances of the tests based on spatial signs and ranks and different tests based on sample means do not depend much on the heaviness of the tails of the distributions. Instead, their performances largely depend on the underlying dependence structure. All these tests have the same asymptotic power under mixing conditions on the coordinate variables as the dimension grows and the sample size is fixed. This is in striking contrast to the performances of these tests in the classical multivariate setup, where the data dimension is smaller than the sample size, and the asymptotic analysis is done by letting the sample size grow while the dimension is held fixed. In some other models involving stronger dependence among the coordinate variables, spatial sign and rank based tests significantly outperform mean based tests.

R V Ramamoorthi

A Proof of Blackwell`s theorem on comparison of experiments

A statistical experiment \(\mathcal{E}\) is a family \(\{P_\theta: \theta \in \Theta\}\) of probability measures on a measurable space \( (\mathcal{X}, \mathcal{A})\). Let \(\mathcal{E}, \mathcal{F}\) be two experiments with \(\Theta\) a finite set. D.Blackwell defined two notions of \( \mathcal{E} \) ` better ' than \(\mathcal{F}\) and showed, using a martingale approximation, that these two notions are equivalent. Beyond the statistical context, this result also yields one of the early dilation theorem on compact convex sets.

In this talk we will give a simpler, essentially decision theoretic proof, of Blackwell's theorem. This talk is based on joint work with D.Feldman

Rajat Shubhra Hazra

Extremes of some Gaussian random interfaces

In this talk we show some supercritical Gaussian interface models belong to maximal domain of attraction of Gumbel, following an application of the Stein-Chen method studied in Arratia et al (1989). We discuss some of the well-known models in statistical mechanics, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field. We give a general sufficient criterion for convergence for such dependent Gaussian fields to Gumbel domain of attraction. The talk is based on joint works with Alberto Chiarini and Alessandra Cipriani.

Shankar Bhamidi

Scaling limits of critical random graph models

Over the last few years, one major theme in the study of random discrete structures, is the notion of scaling limits; these discrete objects properly rescaled converge to continuum random objects in the large size limit. Concurrently, motivated by the presence of empirical data on a wide array of real world networks, there has been an explosion in the number of network models proposed to explain various functionals observed in real world systems including power law degree distribution and small world phenomenon.
Most of these models come with a parameter \(t\) (usually related to edge density) and a (model dependent) critical time \(t_c\) which specifies when a giant connected component (containing a positive faction of the vertices) emerges. A major open problem in this area, since the time of Erdos and Renyi is an understanding of properties of components in the critical regime namely at \(t_c\). In the last decade, based on a number of computational studies, there is mounting evidence to support that for a wide class of models, the nature of this emergence is universal in the sense that:
(a) If the degree distribution of the random graph model has finite third moments then maximal component sizes in the critical regime scale like \(n^{2/3}\) whilst these components scaled by \(n^{-1/3}\) converge to random fractals.
(b) If the degree distribution has finite second but infinite third moments then the sizes of maximal components scale like \(n^a(\tau)\) while these components rescaled by \(n^{-b(\tau)}\) converge to continuum random limiting objects. Here \(a(\tau), b(\tau)\) are explicit functions of the degree exponent.
It is also conjectured that a number of fundamental objects in probabilistic combinatorics including the Minimal spanning tree in the supercritical regime obey the exact same scaling.

We report on recent progress in proving these conjectures. Joint work with Nicolas Broutin, Remco van der Hofstad, Sanchayan Sen and Xuan Wang.

Siva Athreya

The gap between Gromov-vague and Gromov-Hausdorff-vague topology

In talk we define the Gromov-vague topology and the Gromov-Hausdorff-vague topology on the space of metric boundedly finite measure spaces. We explain the necessity of and gap between the two topologies via examples from the literature. This is joint work with Anita Winter and Wolfgang Lohr.

Yogeswaran Dhandapani

Limit theorems for Betti numbers of the Boolean model

We shall look at the classical Boolean model via its topological descriptors called as Betti numbers. In particular, we shall describe strong laws when the underlying point process is an Ergodic point process, concentration inequalities for Binomial and Poisson point processes with 'nice' densities and a central limit theorem for the stationary Poisson point processes. This is a joint work with Eliran Subag and Robert J. Adler.