Probability Seminar

Day, Date and Time : Thursday 26th Sept 3:00 pm

Venue : Ramanujan Hall

Speaker : Parthanil Roy

Title : Detailed Analysis of Phase Transition for Elephant Random Walks
with Two Memory Channels

Abstract : Random processes with strong memory and/or self-excitation
arise naturally in various disciplines including physics, economics,
biology, engineering, geology, etc. Memory can be multifaceted and can
arise due to interactions of more than one underlying phenomena. Many of
these processes exhibit superdiffusive growth due to the effect of
self-excitation. A class of one-dimensional, discrete-time such models
called “elephant random walk with n memory channels” was introduced and
discussed in a recent paper on statistical physics by Saha (2022). In
these models the information of n previous steps from the walker’s entire
history is needed to decide the future step. The aforementioned work
carried out a bunch of calculations, and conjectured a phase transition
from diffusive to superdiffusive regime based on some numerical
computations in the n=2 case. We have been able to prove these conjectures
rigorously and establish a few new transition boundaries beyond the
predicted ones. I shall present these results along with several open
problems. (This talk is based on a joint work with Krishanu Maulik and
Tamojit Sadhukhan.)

7. Algebraic Groups Seminar

Day, Date and Time : Friday 27th Sept, 4:00 pm

Venue : Room 215

Speaker : Akash Yadav

Title : Nilpotent Conjugacy Classes

Abstract : We will continue the analysis of nilpotent conjugacy classes
using the book by Collingwood and McGovern.
 

Annual Progress Seminar

Day, Date and Time : Friday, 27th Sept, 4 pm.

Venue : Ramanujan hall

Speaker : Mayukh Choudhury

Title: Asymptotic Inferences in Generalized Linear Models

Abstract: This talk will be bifurcated into two segments. In the first
part, we will mainly indulge in "Asymptotic Properties of Cross-Validated
Lasso Estimator in GLM". This is precisely the continuation of our last
talk. The penalty parameter in LASSO, is generally chosen in a data
dependent way in practice. Among them, the K-fold CV is the most
celebrated one. So far we have defined the K-fold CV Lasso estimator
$\hat{\lambda}_{n,K}$ and explored its asymptotic properties in terms of
establishing the consistency of the sequence $n^{-1}\hat{\lambda}_{n,K}$
and boundedness of the sequence $n^{-1/2}\hat{\lambda}_{n,K}$ . However,
to justify the distributional convergence of LASSO estimator, we usually
need the convergence of the sequence $n^{-1/2}\hat{\lambda}_{n}$. Towards
that, we will prove the sequence $n^{-1/2}\hat{\lambda}_{n,K}$ is Cauchy
under some additional conditions. Now boundedness together with Cauchy
will serve our purpose. With this we will summarise and conclude this
segment.

In the later half, we will talk about "Large Dimensional CLT in GLM over
Convex Sets and Balls". We will aim to approximate the distribution of
properly centered and scaled GLM estimator with Gaussian random vector
under finite fourth moment condition uniformly over convex sets and
Euclidean Balls precisely when the dimension of the parameter vector, d
can grow with n. For class of measurable convex sets we obtain that d can
grow as o(n^{2/5}) and that for Euclidean Balls, we get d=o(n^{1/2}).
These are the best possible rates that we can have, which are similar to
the findings of Fang and Koike (2024). Lastly, we will prove the Bootstrap
approximation results for the distribution of properly centered and scaled
GLM estimator when the covariance matrix of the Gaussian random vector is
usually unknown.