Poster Presentation

Lectures on Probability and Stochastic Processes XVI

Poster Presenter: Aritra Mandal (ISI Bengaluru, India)

Title: Dynamics of Random Sparse graph limits

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Poster Presenter: Jnaneshwar Baslingker (IISc Bengaluru, India)

Title: Stochastic domination in beta ensembles

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Poster Presenter: Anant Muse (IISc Bengaluru)

Title: Chemical Graph Theory

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Poster Presenter: Sarvesh Iyer (ISI Bengaluru, India)

Title : Elliptic Harnack inequality and Conformal walk dimension of the geometric stable process

Abstract : The parabolic and elliptic Harnack inequalities(abbreviated PHI and EHI respectively) are fundamental regularity estimates that have many applications in differential equations and probability. In an attempt to relate them, Kajino and Murugan defined the conformal walk dimension of a process as the infimum of all $\beta>0$ such that a space-time changed version of the process satisfies $\text{PHI}(\beta)$. They showed that the EHI for symmetric diffusions on metric measure spaces is characterized by the conformal walk dimension being equal to 2. In an attempt to refute this characterization for symmetric jump processes, we define geometric stable processes and show that they satisfy the EHI but have infinite conformal walk dimension. Joint work with Prof. Siva Athreya and Prof. Mathav Murugan.

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Poster Presenter: Atin Gayan (IIT Rorkee, India)

Title: Generalized Rao-Blackwell-type estimators and Cramer-Rao-type bounds for certain power-law distributions

Abstract: This work rethinks statistical sufficiency, moving beyond Fisher's classical formulation. We generalize sufficiency based on the Jones et al. likelihood function, popular in information theory and robust statistics. This leads to certain power-law distributions, like the Student-t, exhibiting a fixed set of sufficient statistics irrespective of sample size. These statistics prove to be the best estimators with respect to a deformed form of the original power-law family. We finally establish a variance bound for these estimators, equals the asymptotic variance of Jones et al. estimator.

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Poster Presenter: Kiran Kumar (IIT Bombay, India)

Title: A unified approach to linear eigenvalue statistics of generalized patterned random matrices

Abstract: A generalized patterned random matrix is defined as $A=(x_{L(i,j)}\mathbf{1}_{\Delta}(i,j))_{i,j=1}^N$ , where the input sequence $\{x_i\}$ is a sequence of independent random variables, $L$ is known as the link function and $\Delta \subseteq \{1,2,\ldots, N\}^2$ . Several important classes of random matrices, including the Wigner Matrix, Toeplitz matrix, Hankel matrix, circulant-type matrices and the block versions of these matrices, fall into this category. In 2006, Bose and Sen showed that under some restrictions on the link function $L$ , the corresponding patterned matrices always have a sub-gaussian limiting spectral distribution.

In this work, we study the linear eigenvalue statistics of $A_n/\sqrt{n}$ , given by

$\eta_p= \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \lambda_i^p,$

where $\lambda_i$ are the eigenvalues of $A_n/\sqrt{n}.$ We show that when $p$ is even and under similar restrictions on $L$ , $\eta_p - \mathbb{E} \eta_p$ converges in distribution either to a normal distribution or to the degenerate distribution at zero as $n \rightarrow \infty$ . We show that under further assumptions on $L$ , the limit is always a normal distribution, and additionally, we show that Toeplitz, Hankel, circulant-type matrices and block versions of these matrices obey these assumptions. We also derive the limiting moments for $\eta_p$ when $p$ is odd.