Here is a list of my research articles along with their abstracts.
Limit Theorems for a class of unbounded observables with an application to “Sampling the Lindelof Hypothesis”
with Tanja Schindler, 49 pp (draft available)Abstract
We prove the Central Limit Theorem, a Mixing Local Limit Theorem and the first order Edgeworth expansion for the Birkhoff sum of a class of $L^3$ observables over Boolean-type transformations on $\mathbb{R}$ using the spectral approach based on the Keller-Liverani perturbation result. The class of observables include the real part, the imaginary part and the absolute value of Riemann zeta function. This result is in the spirit of a result by Steuding who has proven a strong law of large numbers for sampling the Lindelof Hypothesis.
The Bootstrap for dynamical systems
with Nan Zou, 69 pp (submitted)Abstract
Despite their deterministic nature, dynamical systems often exhibit seemingly random behaviour. Consequently, a dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using statistical methods. When measuring the uncertainty of such parameter estimation, the bootstrap stands out as a simple but powerful technique. In this paper, we develop the bootstrap for dynamical systems and establish not only its consistency but also its second-order efficiency via a novel continuous Edgeworth expansion for dynamical systems. Moreover, we verify the theoretical results about the bootstrap using computer simulations.
An error term in the Central Limit Theorem for sums of discrete random variables
with Dmitry Dolgopyat, Internantional Mathematics Research Notices, 42 pp (accepted with revisions)Abstract
We consider sums of independent identically distributed random variables those distributions have $d+1$ atoms. Such distributions never admit an Edgeworth expansion of order $d$ but we show that for almost all parameters the Edgeworth expansion of order $d-1$ is valid and the error of the order $d-1$ Edgeworth expansion is typically of order $n^{-d/2}$.
Robust normalizing flows using Bernstein-type polynomials
with Sameera Ramasinghe, The 33rd British Machine Vision Conference (BMVC), November 2022, London, UK, 19 ppAbstract
Normalizing flows (NFs) are a class of generative models that allows exact density evaluation and sampling. We propose a framework to construct NFs based on increasing triangular maps and Bernstein-type polynomials. Compared to the existing (universal) NF frameworks, our method provides compelling advantages like theoretical upper bounds for the approximation error, robustness, higher interpretability, suitability for compactly supported densities, and the ability to employ higher degree polynomials without training instability. Moreover, we provide a constructive universality proof, which gives analytic expressions of the approximations for known transformations. We conduct a thorough theoretical analysis and empirically demonstrate the efficacy of the proposed technique using experiments on both real-world and synthetic datasets.
Expansions in the Central and the Local Limit Theorems for Dynamical Systems
with Françoise Pène, Communications in Mathematical Physics 389, 273–347, 75 pp (2022) doi:10.1007/s00220-021-04255-zAbstract
We study higher order expansions both in the Berry-Ess&een estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting.
Edgeworth expansions for weakly dependent random variables
with Carlangelo Liverani , Annales de l'Institut Henri Poincaré, 57(1) (2021) 469-505, 37 pp: doi:10.1214/20-AIHP1085Abstract
We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the CLT for weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps and strongly ergodic Markov chains. We primarily use spectral techniques to obtain the results.
Higher order asymptotics for large deviations - Part II
with Pratima Hebbar, Stochastics and Dynamics, 21(5) (2021) 21 pp: doi:10.1142/S0219493721500258Abstract
We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a d-dimensional compact manifold admit these asymptotic expansions of all orders.
Higher order asymptotics for large deviations - Part I
with Pratima Hebbar, Asymptotic Analysis, 121 (2021) 219–257, 39 pp: doi:10.3233/ASY-201602Abstract
For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We apply our results to show that Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type satisfy these strong large deviation results.