Day 1 (December 15)

Speaker: Vivek Shripad Borkar (IIT Bombay)

Title: Asymptotic convexity in shallow and wide networks

Abstract: It is observed that shallow and wide neural networks perform well. We use a stylized model to give a mathematical explanation of this phenomenon. The proof uses a result of Shapley-Folkman-Starr regarding convergence of normalized Minkowski sums of a compact set with itself to its closed convex hull.

Speaker: Subhajit Dutta (ISI Kolkata)

Title: Uniform-over-dimension asymptotic theory with an application to high-dimensional testing of locations

Abstract: Asymptotic methods for hypothesis testing for high-dimensional data usually require the dimension of the observations to increase to infinity, and often involve additional conditions on its rate of increase with the sample size. On the other hand, multivariate asymptotic methods are valid for fixed dimensions only and their practical implementations typically require the sample size to be large compared to the dimension for yielding desirable results. In practice, it is usually not possible to determine whether the dimension of the data at hand conforms to the conditions required for the validity of the high-dimensional asymptotic methods. In this work, we develop a theory of asymptotic convergence which holds uniformly over the dimension of the random vectors. Our proposed theory attempts to unify the asymptotic results for fixed-dimensional multivariate and high-dimensional data. An application of this theory is demonstrated in testing equality of locations for two-sample problems.

Speaker: Alok Goswami (Indian Association for the Cultivation of Science)

Title: On Dual of $L^1$

Abstract: I will present a small yet interesting (and perhaps not yet discovered) observation on the Dual of $L^1$ for arbitrary measure spaces. The observation was made by B.V.Rao and myself, while writing our recently published book on meaure theory.

Speaker: Antar Bandyopadhyay (ISI Delhi)

Title: Interacting Urn Schemes

Abstract: In this talk, we will introduce a novel model of “interactive urn schemes” with the goal of obtaining a limiting distribution, which may be considered as a simple and solvable example of “Self-Organized Criticality (SOC)”. The interactions will be defined via a network (possibly infinite). We will show that limit exists under very minimal condition on the underline graph. [This is a joint work with Deborshi Das]

Speaker: Sayan Ranjan Bhowal (ISI Kolkata)

Title: Statistical inference using debiased group graphical lasso for multiple sparse precision matrices

Abstract: Debiasing group graphical lasso estimates enables statistical inference when multiple Gaussian graphical models share a common sparsity pattern. In this talk, we analyze asymptotic properties of group graphical lasso estimates, establishing convergence rates and model selection consistency under irrepresentability conditions. Based on these results, we construct debiased estimators that are asymptotically Gaussian, allowing hypothesis testing for linear functionals of precision matrices across populations. Performance of the method is examined using synthetic and real-world datasets.

Speaker: Subhodeep Dey (ISI Kolkata)

Title: Cross-Sectional Regression with Cluster Dependence: Inference based on Averaging

Abstract: In this talk, we re-examine the asymptotic properties of the traditional Pooled Ordinary Least Squares (POLS) estimator under cluster dependence. We show that in many realistic scenarios, POLS can be inconsistent. To address this, we propose a simple estimator based on data averaging. The proposed estimator remains consistent even when POLS fails and is often more efficient. As a by-product of its averaging structure, the proposed estimator also retains consistency and asymptotic normality under classical measurement error, thus avoiding reliance on instrumental variables (IV). A comprehensive simulation study illustrates the efficacy of the proposed estimator, and empirical findings suggest that it delivers improved goodness of fit.

Speaker: K. S. Mallikarjuna Rao (IIT Bombay)

Title: Maximum Principle for Infinite Horizon Optimal Control

Abstract: PMP is well studied for finite horizon optimal control. However, the extension to infinite horizon case is not easy. In this talk, we present some issues concerning the maximum principle for infinite horizon optimal control.

Speaker: Suresh Kumar K (IIT Bombay)

Title: Mean field games under degenerate diffusion dynamics

Abstract: We study MFG when the state dynamics is governed by degenerate controlled SDEs.

Speaker: Anuj Bhowmik (ISI Kolkata)

Title: Carathéodory-Type Selection and Random Fixed Point Theorems for Discontinuous Correspondences

Abstract: This paper explores Carathéodory-type selections inspired by applications in economics and game theory. We present new theorems on such selections for correspondences that need not satisfy standard continuity conditions, such as lower or upper semicontinuity. The results extend existing selection theorems and yield several corollaries. Furthermore, we establish new random fixed-point theorems, random maximal element theorems, and results on random equilibria, including both Nash and Bayesian equilibria. These findings broaden and refine several classical results in the field.

Day 2 (December 16)

Speaker: Rohit Allena (University of Houston)

Title: Confident Risk Premiums and Investments using Machine Learning Uncertainties

Abstract: This paper derives ex-ante confidence intervals for stock risk premium forecasts that are based on a wide range of linear and machine learning models. Exploiting the cross-sectional variation in the precision of risk premium forecasts, I provide improved investment strategies. The confident-high-low strategies that take long-short positions exclusively on stocks with precise risk premium forecasts outperform traditional high-low strategies in delivering superior out-of-sample returns and Sharpe ratios across all models. The outperformance increases (decreases) with the model complexity (bias). The confident-high-low strategies are economically interpretable as trading strategies of ambiguity-averse investors who account for confidence intervals around risk premium forecasts.

Speaker: Anindya Goswami (IISER Pune)

Title: Machine Learning in Option Pricing

Abstract: In this talk, I will first revisit theoretical option pricing models quickly. Next, I will present some data-driven approaches for option price prediction whose derivation is based on the no-arbitrage theory of option pricing. The scope and limitations of ML models using the homogeneity hint will be explained. Finally, we derive a common representation space for achieving domain adaptation using the theoretical treatment. The success of implementing this idea is shown using real data. Then we report several experimental results for critically examining the performance of the derived pricing models. The talk is based on two papers I coauthored with my past students, Atharva Tanksale, Sharan Rajani, and Nimit Rana.

Speaker: Pranab Kumar Das (Centre for Studies in Social Sciences Calcutta)

Title: Credit Availability, Firm Investment and Production with Two Sided Incomplete Information

Speaker: The present study aims at an analysis of a two period general equilibrium model of firm investment, production and pricing decisions in a structure with two sided incomplete information. A typical firm’s investment is determined by the availability of bank credit. The firm knows the credit available to itself but does not know what is the availability of credit to other firms. This is one side of the incomplete information. This information is crucial for the price and production decisions of the firm. The banks when lends credit to a typical firm, does not know the exact probability of repayment, but lends on the basis of the industry average. This is another side of the incomplete information set. The decisions of the two sides of the economy, viz. firms and banks interact for the general equilibrium in this model. The decisions of the firms and the banks are significantly different depending on the degree of incompleteness in respect of information. These are also dependent on the market structure in which the firms operate.

Speaker: Aritra Majumdar (ISI Kolkata)

Title: Step reinforced random walks with regularly varying memory

Abstract: The Step Reinforced Random Walk (SRRW) is a generalization of the usual random walk with memory. The walker, at every epoch, chooses a past step uniformly at random; repeats it with probability $p$ (called the recollection parameter) or innovates with probability $1-p$, independent of the past. We study a variant of the SRRW introduced by Bertenghi and Laulin (2024), in which the past step is selected with probability proportional to a weight sequence (called the memory sequence) that is regularly varying of index $\gamma>-1$. In this talk, we analyze the asymptotic behavior of the walk under the assumption that the innovation steps have finite second moments. The phase transition emerging in such a walk is discussed along with the novel superdiffusive scalings appearing in the critical regime. This is a joint work with Dr. Krishanu Maulik.

Speaker: Himasish Talukdar (ISI Kolkata)

Title: Spectra of Contractions of the Gaussian Orthogonal Tensor Ensemble

Abstract: In this talk we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal G$ denote a random tensor of dimension $n$ and order-$r$, drawn from the density

We consider contractions of the form $\mathcal G \cdot \mathbf w^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain a Baik-Ben Arous-Péché phase transition for the largest and the smallest eigenvalues of such contractions at $r = 3$. We also show that the extreme eigenvectors contain non-trivial information about $\mathbf w$. In fact, in the regime $1 \ll r \ll n$, there are two vectors, one of which is perfectly aligned with $w$. We also obtain some results on mixed contractions $\mathcal G \cdot \mathbf u \otimes \mathbf v$ in the case $r = 4$. While the total variation distance between the joint distribution of the entries of $\mathcal G \cdot \mathbf u \otimes \mathbf v$ and that of $\mathcal G \cdot \mathbf u \otimes \mathbf u$ goes to $0$ when $\|\mathbf u - \mathbf v\| = o(n^{-1})$, the bulk and the largest eigenvalues of these matrices have the same limit profile as long as $\|\mathbf u - \mathbf v\| = o(1)$. Further, it turns out that there are no outlier eigenvalues when $\langle \mathbf u, \mathbf v\rangle = o(1)$. This talk is based on a joint work with Soumendu Sundar Mukherjee.

Speaker: Tatiana Turova Schmeling (Lund University)

Title: Phase transitions in Coulomb chains

Speaker: We consider a system of particles lined up on a finite interval in a 3-dimensional space with Coulomb interactions between the nearest and next to the nearest neighbours. The distribution of spacings between the consecutive particles is of interest. The nearest-neighbours interactions case is proved to exhibit multiple phase transitions in the variance of the spacings, depending on the strength of the external force when the number of particles goes to infinity. Assuming zero external force, we show that interactions beyond the nearest ones lead to qualitatively new features of the system. In particular, the order of decay (in terms of the total number of particles) of covariances between the spacings is changed when compared with the former nearest-neighbours case. Furthermore, we discover that the covariances between spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of the number of spacings between them, while their amplitude decays. We present also a conditional Central Limit Theorem for dependent random variables which is derived for our model but can be applied in a more general setting as well.

Speaker: Bilol Banerjee (National University of Singapore)

Title: Minimax Optimal Conditional Two Sample Test via Distance Profiles

Abstract: We study the problem of comparing the conditional distributions of two random variables X and Y given a confounding variable Z, where all variables take values in general metric spaces. We introduce a new discrepancy measure that quantifies conditional distributional differences through joint distance profiles and develop a consistent estimator of this quantity. Building on this estimator, we construct a conditional two-sample test that avoids local smoothing and, therefore, remains simple, broadly applicable, and powerful against general alternatives. We further propose an algorithm to calibrate our test, which is both Pitman efficient and minimax-rate optimal against a large class of alternatives. Through simulations, we demonstrate that the proposed method outperforms existing approaches, and we illustrate its practical utility by comparing the distributions of outputs produced by different large language models using the same prompts.

(This is a joint work with Paromita Dubey, USC Marshall)

Speaker: Krishanu Maulik (ISI Kolkata)

Title: Elephant Random Walk with Two Memory Channels

Abstract: Random processes with strong memory arise naturally in various disciplines including physics, economics, biology, geology, etc. Elephant random walk was introduced by Schutz and Trimper (2004) to study the effect of memory on random walks. It is a special type of random walk that incorporates the information of one randomly chosen past step to determine the future step. It has drawn attention of the Statistical Physics literature as it exhibits superdiffusive growth due to the effect of self-excitation. However, memory of a process can be multifaceted and can arise due to interactions of more than one underlying phenomena. Towards this, Random Walk with $k$ Memory Channels was introduced by Saha (2022), where the information of $k$ randomly chosen past steps is needed to decide the future step. The aforementioned work carried out heuristic calculations of variance, and conjectured phase transitions from diffusive to superdiffusive and from superdiffusive to ballistic regimes in the $k = 2$ case. We have proved these conjectures rigorously (with mild corrections), and discovered a new regime at one of the transition boundaries. In this talk, we shall present these results along with a detailed analysis of the asymptotic behaviour of the walk at different regimes. This talk is based on a joint work with Parthanil Roy and Tamojit Sadhukhan.

Speaker: Debapratim Banerjee (Ashoka University)

Title: Overlap distribution of spherical spin glass models with general eigenvalue distribution of the interaction matrix

Abstract: We show that the replica symmetry of the Gibbs measure of spherical spin systems is a property of the eigenvalue spacing at the edge of the interaction matrix. In particular, our interaction matrix has two large outlier eigenvalues with mutual distance $\frac{c}{n}$. The empirical measure of the rest of the eigenvalues is close to the semicircular law with some rigidity conditions. We prove that in this scenario the overlap distribution of two independent samples from the Gibbs measure has a continuous density at a low enough temperature. Hence, the model is a full replica symmetry-breaking model. One might compare this result with only one outlier eigenvalue. This model comes for the Sherrington-Kirkpatrick model with Curie-Weiss interaction in the ferromagnetic case. Here, it is well known that the model is replica symmetric, although the free energy limit of this model is the same as the free energy limit of our model. In our limited understanding, we believe that this kind of phenomenon cannot be explained by the Parisi approach.