arXiv subjects

Aparna Rajput

Publications and source records attributed to Aparna Rajput.

4 recordsLinked to original sources

arXiv2026

Empirical Transfer Operators and Finite-Sample Change Detection for Noisy Expanding Interval Maps

We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution $\widehat{\pi}_{0,\rho}$. For each later sliding window, we compute $\widehat{\pi}_{t,\rho}$ and define the score \[ S_t=\|\widehat{\pi}_{t,\rho}-\widehat{\pi}_{0,\rho}\|_1. \] Large values of $S_t$ indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.

stat.ML↗

arXiv2026

Stochastic Stability of ACIMs for Piecewise Expanding $C^{1+\varepsilon}$ Maps

We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\varepsilon}$ maps of the interval. For maps $\tau$ in the class $\mathcal{T}([0,1]; s, \varepsilon)$, we consider perturbed Frobenius--Perron operators $P_\delta = Q_\delta P_\tau$, where $Q_\delta$ is a Markov smoothing operator modeling noise of intensity $\delta > 0$. In the generalized bounded variation space $BV_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $\delta$. Consequently, each $P_\delta$ admits an invariant density $h_\delta \in BV_{1,1/p}$, and $h_\delta \to h$ in $L^1$ as $\delta \to 0$, where $h$ is the ACIM density of $P_\tau$. Our proof combines the $BV_{1,1/p}$ framework, adapted from recent ACIM existence results, with uniform quasi-compactness and perturbation theory for transfer operators. This establishes stochastic stability under minimal $C^{1+\varepsilon}$ regularity ($\varepsilon > 0$), where the $C^1$ case is known to fail.

math.DS↗

arXiv2024

Existence of ACIM for Piecewise Expanding $C^{1+\varepsilon}$ maps

In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron Operator of a piecewise expanding $C^{1+\varepsilon}$ map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem \cite{ionescu1950}, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we prove the quasi-compactness of the Frobenius-Perron operator induced by the map. Additionally, we explore significant properties of the system, including weak mixing and exponential decay of correlations.

math.DS↗

arXiv2024

Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches

In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $\tau$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $\tau$, $\tau^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $\mu$ for $\tau$, the exactness of the dynamical system $(I, \tau,\mu)$ and the quasi-compactness of Frobenius-Perron operator $P_\tau$ induced by $\tau$. The last fact implies a multitude of strong ergodic properties of $\tau$.

math.DS↗