Theoretical Statistics and Mathematics Unit

## Seminar and Colloquium of the week

### BANGALORE PROBABILITY SEMINAR

Title : Spectra of Adjacency and Laplacian Matrices of inhomogeneous Erdos-Renyi Graphs
Speaker : Arijit Chakrabarty (ISI Kolkatta)
Time : November 12, 2018 (Monday), 02:00 PM
Venue : G23
Abstract : Inhomogeneous Erd\H{o}s-R\'enyi random graphs $\mathbb{G}_N$ on $N$ vertices in the non-dense regime are studied. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N f(\tfrac{i}{N},\tfrac{j}{N})$, $1 \leq i,j \leq N$, independently of other edges, where $f\colon\,[0,1]\times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $\Delta_N$ associated with $\mathbb{G}_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, we show that the empirical distributions of $(N\varepsilon_N)^{-1/2} A_N$ and $(N\varepsilon_N)^{-1/2} \Delta_N$ converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\, [0,1] \to [0,\infty)$ a continuous function, we give an explicit characterisation of the limiting distributions. \\ \noindent \textbf{Authors:} Arijit Chakrabarty, Rajat Subhra Hazra, Frank den Hollander, and Matteo Sfragara

### COLLOQUIUM

Title :Multivariate Output Analysis for Markov Chain Monte Carlo
Speaker : Dootika Vats (University of Warwick)
Time : November 13, 2018 (Tuesday), 02:00 PM
Venue : EAU Seminar Hall
Abstract : Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are then demonstrated through simple motivating example. This work is joint with Galin Jones (U of Minnesota) and James Flegal (UC Riverside).

### COLLOQUIUM

Title :Boundary behavior of optimal approximants
Speaker : Daniel Seco (ICMAT - Madrid)
Time : November 15, 2018 (Thursday), 03:30 PM
Venue : G23
Abstract : We compute the Taylor coefficients of $p_{nf} -1$, where $p_n$ denotes the optimal approximant of degree $n$ to $1/f$ in a Hilbert space of analytic functions over the unit disc $D$, and $f$ is a polynomial of degree $d$ with $d$ simple zeros. As an application, we show that the sequence $p_{nf}-1$ is uniformly bounded and, if $f$ has no zeros inside the disc, the values of $p_{nf} - 1$ converge locally uniformly towards 0 at every point of the boundary except the zeros of $f$.

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