Related papers: Scaling limits for exploration algorithms
We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of…
We analyse the problem of estimating a scalar parameter $g$ that controls the Hamiltonian of a quantum system subject to Markovian noise. Specifically, we place bounds on the growth rate of the quantum Fisher information with respect to…
We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval $[0,T]$ in the limit $T \rightarrow \infty$. We further exhibit the asymptotic behaviour of the…
The goal of these lectures is to survey some of the recent progress on the description of large-scale structure of random trees. We use the framework of Markov-Branching sequences of trees and discuss several applications.
We consider a multilevel continuous time Markov chain $X(s;N) = (X_i^j(s;N): 1 \leq i \leq j \leq N)$, which is defined by means of Jack symmetric functions and forms a certain discretization of the multilevel Dyson Brownian motion. The…
In this note we consider a Markov chain formed by a finite system of interacting birth-and-death processes on a finite state space. We study an asymptotic behaviour of the Markov chain as its state space becomes large. In particular, we…
We propose a safe exploration algorithm for deterministic Markov Decision Processes with unknown transition models. Our algorithm guarantees safety by leveraging Lipschitz-continuity to ensure that no unsafe states are visited during…
We study a hydrodynamic limit approach to move-to-front rules, namely, a scaling limit as the number of items tends to infinity, of the joint distribution of jump rate and position of items. As an application of the limit formula, we…
Computing the size of maximum independent sets is a NP-hard problem for fixed graphs. Characterizing and designing efficient algorithms to estimate this independence number for random graphs are notoriously difficult and still largely open…
We establish general sufficient conditions for a sequence of controlled branching processes to converge weakly on the Skorokhod space. We focus on a class of controlled random variables that extends previous results by considering them as a…
Stochastic processes find applications in modelling systems in a variety of disciplines. A large number of stochastic models considered are Markovian in nature. It is often observed that higher order Markov processes can model the data…
We give conditions to prove the existence of an Extremal Index for general stationary stochastic processes by detecting the presence of one or more underlying periodic phenomena. This theory, besides giving general useful tools to identify…
For each $n\geq 1$, let $ {X_{in}, \quad i \geq 1} $ be independent copies of a nonnegative continuous stochastic process $X_{n}=(X_n(t))_{t\in T}$ indexed by a compact metric space $T$. We are interested in the process of partial maxima…
Language models famously improve under a smooth scaling law, but some specific capabilities exhibit sudden breakthroughs in performance. Advocates of "emergence" view these capabilities as unlocked at a specific scale, but others attribute…
The article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster and then we apply the developed theory to determine the limiting…
Markov models are widely used to describe processes of stochastic dynamics. Here, we show that Markov models are a natural consequence of the dynamical principle of Maximum Caliber. First, we show that when there are different possible…
We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map (CRUM) of a given genus that start with a suitably tilted Brownian continuum…
A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition…
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently a stationary stochastic process (time…
We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen…