arXiv subjects

Deborshi Das

Publications and source records attributed to Deborshi Das.

3 recordsLinked to original sources

arXiv2026

Interacting Urn Schemes on Infinite Graphs: A Representation Approach

Interacting urn schemes have emerged as an important framework for studying reinforcement phenomena in interacting stochastic systems (see, e.g., \cite{launay2011interacting,launay2012generalized,WARM2014,aletti2017interacting,Synchronisation-2,kaur2023interacting,kaur2023interacting2}). Most existing results, however, are restricted to finite systems. In this article, we investigate interacting urn schemes involving infinitely many urns, where the interactions are naturally described by a directed graph. The reinforcement or drawing mechanism at each urn depends on the compositions of its in-neighboring urns. To analyze the asymptotic behavior of these systems, we develop a representation of the urn dynamics in terms of branching Markov processes evolving on a class of random forests introduced in this article. This extends the representation approach of \cite{bandyopadhyay2022new}, originally developed for single Pólya urn schemes, to interacting urn schemes on locally finite directed graphs. Using this representation, we establish almost sure convergence of the color proportions under standard conditions on the reinforcement matrix or kernel. In particular, we show that the convergence holds independently of the topology of the underlying graph, and hence our results also apply to undirected graphs, where the interactions are bidirectional. We further investigate measure-valued generalizations and other related reinforced stochastic processes, illustrating the broad applicability of the representation approach. Finally, we discuss connections with self-organized criticality.

math.PR↗

arXiv2026

Interacting Urn Schemes on Finite Ancestral Directed Acyclic Graphs

We study interacting finite-color urn schemes on directed acyclic graphs, allowing the graph to be infinite. Each urn evolves through reinforcements driven by colors drawn from its in-neighboring urns via edge-dependent reinforcement matrices. Assuming that every vertex has only finitely many ancestors, we prove almost sure convergence of urn proportions and show that the limiting configuration is determined by vertices with no ancestors or self-loops. Under additional balance and irreducibility assumptions on reinforcement matrices, we also obtain second-order asymptotic results in all regimes of appropriately defined parameters.

math.PR↗

arXiv2024

Elephant random walks with graph based shared memory: First and second order asymptotics

We consider a generalization of the so-called elephant random walk by introducing multiple elephants moving along the integer line, $\mathbb{Z}$. When taking a new step, each elephant considers not only its own previous steps but also the past steps of other elephants. The dynamics of "who follows whom" are governed by a directed graph, where each vertex represents an elephant, and the edges indicate that an elephant will consider the past steps of its in-neighbour elephants when deciding its next move. In other words, this model involves a collection of reinforced random walks evolving through graph-based interactions. We briefly investigate the first- and second-order asymptotic behaviour of the joint walks and establish connections with other network-based reinforced stochastic processes studied in the literature. We show that the joint walk can be expressed as a stochastic approximation scheme. In certain regimes, we employ tools from stochastic approximation theory to derive the asymptotic properties of the joint walks. Additionally, in a specific regime, we use better techniques to establish a strong invariance principle and a central limit theorem with improved rates compared to existing results in the stochastic approximation literature. These techniques can also be used to strengthen equivalent results in stochastic approximation theory. As a byproduct, we establish a strong invariance principle for the simple elephant random walk with significantly improved rates.

math.PR↗