Lecture Abstracts

Andrey Zaitsev, Steklov Mathematical Institute, St. Petersburg

The accuracy of strong Gaussian approximation for sums of random vectors

We discuss the results on the rate of strong approximation in the multidimensional invariance principle for sums of independent random vectors, which were published in the recent papers of Zaitsev and G\(\ddot{\text{o}}\)tze and Zaitsev. They may be considered as multidimensional generalizations and improvements of some results of Koml\(\acute{\text{o}}\)s, Major and Tusn\(\acute{\text{a}}\)dy (1975), Sakhanenko (1985) and Einmahl (1989).

Anish Sarkar, ISI Delhi

Web in the Scaling Limit of Supercritical Oriented Percolation in Dimension \(1+1\)

We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice \(\{(x,i)\in {\mathbb Z}^2: x+i \mbox{ is even}\}\) converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.

Arup Bose, ISI Kolkata

High dimensional time series model and free probability

Consider a high dimensional linear time series model. The corresponding sample autocovariance matrices are crucial objects in any inference procedure. We show that the limiting spectral distribution of symmetrized versions of these matrices exist and also show how to extend this to joint convergence. Ideas from free probability come up naturally in this study and the limits may be described in terms of suitable free independent random elements. Explicit description of the limit is given in some special cases. This work is joint with Monika Bhattacharjee.

Bimal K. Roy, ISI, Kolkata

A Semi-Parametric Software Reliability Model for Modern Bug Database

The Internet has enabled a completely new approach to software Testing: Crowd-sourcing, where users of a software, at a global scale, use a software product and report it's defects in a voluntary and uncontrolled manner through the internet. Crowd-sourced testing is a cost effective strategy for software quality improvement and is an increasingly popular method in the Industry. However due to it's uncontrolled and voluntary nature, almost no existing model for software reliability can be used to make reliability inferences. I will talk about new procedures we have built to address this important problem.

Debasis Kundu, IIT Kanpur

On two different signal processing models

In this talk we will discuss about two different signal processing models, namely (a) Sum of sinusoidal model and (b) Chirp model. Both the models have several applications in different areas. Both the models have received significant attention in the Statistical Signal Processing literature. We will be discussing two new estimators and discuss their properties.

Dimitrii Zaporozhets, Steklov Mathematical Institute, St. Petersburg

Intrinsic volumes and Gaussian processes

Georgy Shevlyakov, St.Petersburg State Polytechnic University

Robust estimation of a correlation coefficient: approaches and methods

Various groups of robust estimators of the correlation coefficient are introduced, including a new family of M-estimators of the correlation coefficient for bivariate independent component distributions. Consistency and asymptotic normality of these estimators are established, and the explicit formulas for their asymptotic variances and biases are obtained. A minimax variance and bias (in the Huber sense) estimator of the correlation coefficient for contaminated bivariate normal distributions is designed. The performance of most prospective estimators is examined at contaminated normal distributions both on small and large samples, and the best of the proposed robust estimators are revealed.

Ildar Ibragimov, Steklov Mathematical Institute, St. Petersburg

Some aspects of non-parametric estimation theory

Isha Dewan, ISI Delhi

On Testing For Equality Of Two Cause Specific Hazard Rates

We consider a series system where one observes the failure time of the system \((T)\) and an indicator function \(\delta\) which identifies the component which leads to failure of the system. Based on random sample of \(T\)'s and \(\delta\)'s, we propose tests for the equality of cause-specific hazard rates against the alternative that they are ordered. Failure times are always observable. But the indicator functions may be observable or could be masked.

K R Parthasarathy, ISI Delhi

Quantum Gaussian Markov Processes

Following the notion of a Gaussian state as outlined in the book of Holevo (1982) and using the definition and properties of Gaussian channels of quantum information theory as outlined in Heinosaari, Holevo and Wolf (2010) we shall present the exact quantum stochastic differential equation obeyed by a quantum Gaussian Markov process with a preassigned covariance structure. Our main tool is the quantum Ito's formula of boson stochastic calculus of Hudson and Parthasarathy (1984).

Ksenia Volkova, St. Petersburg State University

Asymptotic efficiency of goodnness-of-fit tests for the Pareto distributions based on the order statistics characterization

A new characterization of Pareto distribution is proposed, and new goodness-of-fit tests based on it are constructed. Tests statistics are functionals of U-empirical processes. The first of this statistics is of integral type, the second one is a Kolmogorov type statistic. We show that the kernels of our statistics is non-degenerate. The limiting distribution and large deviations asymptotic of new statistics under null hypothesis are described. Their local Bahadur efficiency for parametric alternatives is calculated. This type of efficiency is mostly appropriate for solution of our problem since the Kolmogorov type statistic is not asymptotically normal, and the Pitman approach is not applicable to this statistic. For second statistic we evaluate the critical values by using Monte-Carlo methods. Also conditions of local optimality of new statistics in the sense of Bahadur are discussed and examples of such special alternatives are given.

Kumarjit Saha, ISI Delhi

Some quantitative estimates in a drainage network model

Hack, [1957] while studying the drainage system in the Shenandoah valley and the adjacent mountains of Virginia observed a power law relation \(L = 1.4 A^{0.6}\) between the length \(L\) of a stream from its source to a divide and the watershed area \(A\) formed by its tributaries upstream from the divide. In this work we study the tributary structure of a 2-dimensional drainage network model known as Howard's model of headward growth and branching studied by Gangopadyay, Roy and Sarkar [2004]. Our study is based on a scaling of the process and we obtain the watershed area of a stream as the area of a Brownian excursion process. This gives a statistical explanation of Hack's law and justifies the remark of Giacometti et al. ``From the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of Hack's law.'' The proof uses the convergence of drainage network to the Brownian web under suitable scaling (Coletti, Fontes, Dias [2009]). We define a dual system and show that the drainage network and the dual jointly converge in distribution to the Brownian web and its dual. The method of the proof is quite general and can be used for other drainage network models also which under diffusive scaling converge to the Brownian web.

This is a joint work with Professor Rahul Roy and Professor Anish Sarkar.

Manjunath Krishnapur, IISc Bengaluru

Some anti-concentration inequalities

Natalia Smorodina, St. Petersburg State University

Limit theorems for sums of independent random variables and Feynman type integrals

We discuss a possibility to construct both a probabilistic representation and a probabilistic approximation of the Cauchy problem solution for an equation \(\frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\,\Delta u+V(x)u,\) where \(\sigma\) is a complex parameter such that \(\mathrm{Re}\,\sigma^2\geqslant 0\). This equation coincides with the heat equation when \(\mathrm{Im}\,\sigma=0\) and with the Schr\(\ddot{\text{o}}\)dinger equation when \(\mathrm{Re}\,\sigma^2=0\).

Oleg Rusakov, St. Petersburg State University

Sums of Pseudo-Poisson processes with random intensities; applications to the stochastic models of spot-rates

Olga Klopp, University Paris Ouest, France

1-bit Matrix Completion

Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of interest. This model corresponds, for example, to ratings in recommender systems or labels in multi-class classification. This is a joint work with Jean Lafond, Eric Moulines et Joseph Salmon.

Parthanil Roy, ISI Kolkata

Stable Random Fields, Point Processes and Large Deviations

We investigate the large deviation behaviour of a point process sequence based on a stationary symmetric stable non-Gaussian random field using the framework of Hult and Samorodnitsky (2010). Depending on the ergodic theoretic and group theoretic structures of the underlying nonsingular group action, we observe different large deviation behaviours of this point process sequence. We use our results to study the large deviations of various functionals (e.g., partial sum, maxima, etc.) of stationary symmetric stable fields. This talk will be based on a joint work with Vicky Fasen.

Probal Chaudhuri, ISI Kolkata

Nonparametric Statistics in Dimensions One, Two, Three, \(\ldots\) , Infinity

Various extensions of classical nonparametric statistical procedures applicable to finite dimensional multivariate data have been extensively studied in the past. There are many different versions of ranks, quantiles and order statistics for multivariate data available in the literature. However, when the observed data lies in an infinite dimensional space, many of these finite dimensional versions of the basic nonparametric statistical tools cease to have meaningful extensions that can handle such data. I shall discuss some of the fundamental difficulties in nonparametric statistical analysis of infinite dimensional data and how to cope with the challenge.

Rajesh Sundaresan, IISc Bengaluru

Information geometry of the robust maximum mean power likelihood estimation

The maximum "mean power" likelihood estimation procedure (proposed by [Basu et al., Biometrika, 1998]) is a robust variant of the maximum likelihood estimation procedure. Given iid samples coming from an unknown distribution, a member of a given parametric family of distributions, the mean power likelihood estimation procedure is to find the parameter of the distribution that is closest, in a particular sense, to the empirical distribution of the samples. One may view the resulting distribution as a "projection" of the empirical distribution on the parametric family. In this talk, I will highlight the geometry associated with this projection. I will also discuss a simplified computation procedure, one that is suggested by the geometric view point, when the estimation is of the parameter of a power-law family.

Siva Athreya, ISI Bengaluru

Invariance principle for variable speed random walks on trees

We consider stochastic processes on complete, locally compact tree-like metric spaces \((T,r)\) on their "natural scale" with boundedly finite speed measure \(\nu.\) In this talk we shall show that speed-\(\nu_n\) motions on \((T_n,r_n)\) converge weakly in path space to the speed-\(\nu\) motion on \((T,r)\) provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology.

Vivek Borkar, IIT Mumbai

Q-learning without stochastic approximation

This talk will describe a novel scheme for approximate dynamic programming for Markov decision processes using simulated data, but which, unlike conventional reinforcement learning schemes, is not based on stochastic approximation and therefore exhibits qualitatively different behaviour. (Joint work with Dileep Kalathil (UC, Berkeley) and Rahul Jain (USC)).

Vladislav Vysotsky, Steklov Mathematical Institute, St. Petersburg

On gaps and non-visited sites within the range of a random walk

Yakov Nikitin, St. Petersburg State University

U-max statistics and asymptotic behavior of random polygons

Recently W. Lao and M. Mayer began the study of U-max - statistics, where instead of the sum appears the maximum over the same set of indices. Such statistics often appear in stochastic geometry. Their limit distribution is related to the distribution of extreme values. Among the interesting results obtained by Lao and Mayer are limit theorems for the maximal perimeter and the maximal area of random triangles inscribed in a circumference. In the present paper we generalize these theorems to convex m- polygons, m > 2, with random vertices on the circumference. Next, a similar problem is solved for the minimal perimeter and the minimal area of circumscribed m-polygons. Finally, we discuss possible generalizations and the case of growing number of vertices.