arXiv subjects

Chinmoy Bhattacharjee

Publications and source records attributed to Chinmoy Bhattacharjee.

At least 19 recordsLinked to original sources

arXiv2026

Dickman Approximation of Randomly Weighted Sums via Stein's Method

We introduce a unified framework via Stein's method for bounding the Kolmogorov distance between the generalized Dickman distributions and the distribution of randomly weighted sums of non-negative integer-valued random variables that are conditionally independent given the weights. By utilizing size-bias couplings and a decomposition of the solution to the corresponding Stein equation into distinct bounded and non-increasing components, our approach yields non-asymptotic error bounds governed only by the quality of a discrete-to-continuous coupling. A similar bound is also obtained for the smooth Wasserstein-2 distance. We apply our abstract results to establish concrete, and in some cases optimal, rates of convergence across diverse probabilistic models. These applications range from settings with independent weights, such as randomly weighted sums of log-primes, to models with intricate dependency structures, including geometric sums over the spectra of the Circular Unitary Ensemble (CUE) and random weights generated by independent increments.

math.PR↗

arXiv2026

Perimeter length of the convex hull of Brownian motion in the hyperbolic plane

We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian motion, and hence derive small- and large-time asymptotics for the expected hyperbolic perimeter. In contrast to the case of Euclidean Brownian motion with non-zero drift, the large-time asymptotics are a factor of two greater than the lower bound implied by the fact that the convex hull includes the hyperbolic line segment from the origin to the endpoint of the hyperbolic Brownian motion. We also obtain an exact expression for the expected perimeter length after an independent exponential random time.

math.PR↗

arXiv2026

Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry

We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincaré inequality is suboptimal, a phenomenon known as superconcentration. In this paper, we establish a rigorous mathematical equivalence between superconcentration and the chaotic behaviour of the functional, and certain associated random sets, under perturbations driven by the Ornstein-Uhlenbeck semigroup on the Poisson space. Leveraging the Malliavin-Stein method, we develop general variance identities and bounds for Poisson functionals, providing a unified framework to prove superconcentration, particularly for geometric functionals that can be expressed as a sum of local score functions. We apply our results to rigorously establish superconcentration and the chaotic behaviour in some models of stochastic geometry. Specifically, we analyse horizontal box-crossing indicators in certain critical continuum percolations, as well as the number of vertices with small degrees and the number of isolated $Γ$-components in random geometric graphs in the dense regime.

math.PR↗

arXiv2025

Gaussian approximation for Extreme Points in Laguerre tessellations

We consider Gaussian approximation in three particular models of Poisson-Laguerre tessellations, namely, the $β$-, $β'$- and Gaussian-Voronoi tessellations. The tessellations are constructed based on inhomogeneous Poisson point processes in space-time $\mathbb{R}^d \times \mathbb{R}$, where some of the points of the process give rise to a cell in $\mathbb{R}^d$, known as extreme points, while the other points produce an empty cell. Using the notion of region-stabilization, we derive quantitative central limit theorems with presumably optimal rates of convergence for the number of extreme points of $β$-, $β'$- and Gaussian-Voronoi tessellations in a growing window $W_n=[-n,n]^d$ as $n\to\infty$. Our bounds improve and extend previously known results by Schreiber and Yukich (2008) for the $β$-model, and are the first quantitative results for the $β'$- and Gaussian models.

math.PR↗

arXiv2025

Multivariate Gaussian Approximation for Random Forest via Region-based Stabilization

We derive Gaussian approximation bounds for $k$-Potential Nearest Neighbor ($k$-PNN) based random forest predictions based on a set of training points given by a Poisson process under fairly mild regularity assumptions on the data generating process. Our approach is based on the key observation that $k$-PNN based random forest predictions satisfy a certain geometric property called region-based stabilization. We also compare the rates with those of $k$-nearest neighbor-based random forests, highlighting a form of universality in our result. In the process of developing our results, we also establish a probabilistic result on multivariate Gaussian approximation bounds for general functionals of Poisson process that are region-based stabilizing. This general result makes use of the Malliavin-Stein method, and is potentially applicable to various related statistical problems.

math.ST↗

arXiv2024

Gaussian and Bootstrap Approximation for Matching-based Average Treatment Effect Estimators

We establish Gaussian approximation bounds for covariate and rank-matching-based Average Treatment Effect (ATE) estimators. By analyzing these estimators through the lens of stabilization theory, we employ the Malliavin-Stein method to derive our results. Our bounds precisely quantify the impact of key problem parameters, including the number of matches and treatment balance, on the accuracy of the Gaussian approximation. Additionally, we develop multiplier bootstrap procedures to estimate the limiting distribution in a fully data-driven manner, and we leverage the derived Gaussian approximation results to further obtain bootstrap approximation bounds. Our work not only introduces a novel theoretical framework for commonly used ATE estimators, but also provides data-driven methods for constructing non-asymptotically valid confidence intervals.

math.ST↗

arXiv2024

Spectra of Poisson functionals and applications in continuum percolation

Let $η$ be a Poisson random measure (defined on some Polish space), and let $F(η)$ be a square-integrable functional of $η$. In this paper we define and study a new notion of {\it spectral point process} associated with $F(η)$, and use such an object to study sharp noise instability and sensitivity properties of planar critical continuum percolation models under spatial birth-death (Ornstein-Uhlenbeck) dynamics -- the notion of sharp noise instability being a natural strengthening of the absence of noise stability. The concept of spectral point process is defined by exploiting the Wiener-Itô chaos expansion of $F$, and represents a natural continuum counterpart to the notion of {\it spectral sample}, as introduced in Garban, Pete and Schramm (2010), in the context of discrete percolation models. In the particular case where $η$ is a marked Poisson measure, we use Hoeffding-ANOVA decompositions to establish an explicit connection with the notion of {\it annealed spectral sample}, introduced in Vanneuville (2021) in the context of Poisson-Voronoi percolation. We also relate spectral processes with an appropriate notion of {\it pivotal processes}. As applications, we show sharp noise instability of crossing events in the critical Poisson Boolean model with unit-radius balls and, using an observation of Vanneuville, we obtain sharp noise sensitivity (as well as sharp noise instability) for crossing events in the Poisson Voronoi percolation model. As an important ingredient, we prove quasi-multiplicativity of the $4$-arm probabilities in the critical Poisson Boolean percolation model.

math.PR↗

arXiv2023

Central limit theorem for a birth-growth model with Poisson arrivals and random growth speed

We consider Gaussian approximation in a variant of the classical Johnson--Mehl birth-growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution and a weight function $h:\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

math.PR↗

arXiv2022

Gaussian approximation for rooted edges in a random minimal directed spanning tree

We study the total $α$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in Bhatt and Roy (2004) - on a Poisson process with intensity $s \ge 1$ on the unit cube $[0,1]^d$ for $d \ge 3$. While a Dickman limit was proved in Penrose and Wade (2004) in the case of $d=2$, in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when $α=1$, with a rate of convergence of the order $(\log s)^{-(d-2)/4} (\log \log s)^{(d+1)/2}$. In this paper, we extend these results and prove a central limit theorem in any dimension $d \ge 3$ for any $α>0$. Moreover, making use of recent results in Stein's method for region-stabilizing functionals, we provide presumably optimal non-asymptotic bounds of the order $(\log s)^{-(d-2)/2}$ on the Wasserstein and the Kolmogorov distances between the distribution of the total $α$-powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.

math.PR↗

arXiv2022

Dickman Approximation of weighted random sums in the Kolmogorov distance

We consider distributional approximation by generalized Dickman distributions, which appear in number theory, perpetuities, logarithmic combinatorial structures and many other areas. We prove bounds in the Kolmogorov distance for the approximation of certain weighted sums of Bernoulli and Poisson random variables by members of this family. While such results have previously been shown in Bhattacharjee and Goldstein (2019) for distances based on smoother test functions and for a special case of the random variables considered in this paper, results in the Kolmogorov distance are new. We also establish optimality of our rates of convergence by deriving lower bounds. As a result, some interesting phase transitions emerge depending on the choice of the underlying parameters. The proofs of our results mainly rely on the use of Stein's method. In particular, we study the solutions of the Stein equation corresponding to the test functions associated to the Kolmogorov distance, and establish their smoothness properties. As applications, we study the runtime of the Quickselect algorithm and the weighted depth in randomly grown simple increasing trees.

math.PR↗

arXiv2022

Gaussian approximation for sums of region-stabilizing scores

We consider the Gaussian approximation for functionals of a Poisson process that are expressible as sums of region-stabilizing (determined by the points of the process within some specified regions) score functions and provide a bound on the rate of convergence in the Wasserstein and the Kolmogorov distances. While such results have previously been shown in Lachièze-Rey, Schulte and Yukich (2019), we extend the applicability by relaxing some conditions assumed there and provide further insight into the results. This is achieved by working with stabilization regions that may differ from balls of random radii commonly used in the literature concerning stabilizing functionals. We also allow for non-diffuse intensity measures and unbounded scores, which are useful in some applications. As our main application, we consider the Gaussian approximation of number of minimal points in a homogeneous Poisson process in $[0,1]^d$ with $d \ge 2$, and provide a presumably optimal rate of convergence.

math.PR↗

arXiv2021

Large degrees in scale-free inhomogeneous random graphs

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution.

math.PR↗

arXiv2020

On Beltrami states near black hole event horizon

In this article, we study Beltrami equilibria for plasmas in near the horizon of a spinning black hole, and develop a framework for constructing the magnetic field profile in the near horizon limit for Clebsch flows in the single-fluid approximation. We find that the horizon profile for the magnetic field is shown to satisfy a system of first-order coupled ODEs dependent on a boundary condition for the magnetic field. For states in which the generalized vorticity vanishes (the generalized `superconducting' plasma state), the horizon profile becomes independent of the boundary condition, and depend only on the thermal properties of the plasma. Our analysis makes use of the full form for the time-independent Ampere's law in the 3+1 formalism, generalizing earlier conclusions for the case of vanishing vorticity, namely the complete magnetic field expulsion near the equator of an axisymmetric black horizon assuming that the thermal properties of the plasma are symmetric about the equatorial plane. For the general case, we find and discuss additional conditions required for the expulsion of magnetic fields at given points on the black hole horizon. We perform a length scale analysis which indicates the emergence of two distinct length scales characterizing the magnetic field variation and strength of the Beltrami term, respectively.

gr-qc↗

arXiv2020

Convergence to scale-invariant Poisson processes and applications in Dickman approximation

We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence $(z_n)_{n\in\mathbb{N}}$ of positive real numbers increasing to infinity as $n \to \infty$ and a sequence $(X_k)_{k\in\mathbb{N}}$ of independent non-negative integer-valued random variables, we consider the sequence of point processes \begin{equation*} ν_n=\sum_{k=1}^\infty X_k δ_{z_k/z_n}, \quad n\in \mathbb{N}, \end{equation*} and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process $η_c$ on $(0,\infty)$ with the intensity measure having the density $ct^{-1}$, $t\in(0,\infty)$. An important motivating example from probabilistic number theory relies on choosing $X_k \sim {\rm Geom}(1-1/p_k)$ and $z_k=\log p_k$, $k\in \mathbb{N}$, where $(p_k)_{k \in \mathbb{N}}$ is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals $\int_0^1 t ν_n(dt)$ to the integral $\int_0^1 t η_c(dt)$, the latter having a generalized Dickman distribution, thus providing a new way of proving Dickman convergence results. We extend our results to the multivariate setting and provide sufficient conditions for vague convergence in distribution for a broad class of sequences of point processes obtained by mapping the points from $(0,\infty)$ to $\mathbb{R}^d$ via multiplication by i.i.d. random vectors. In addition, we introduce a new class of multivariate Dickman distributions which naturally extends the univariate setting.

math.PR↗

arXiv2019

Black hole in a superconducting plasma

The generalized vortical formalism provides an electrodynamic description for superconducting states---in the generalized vortical formalism, a superconducting state may be defined by the vanishing of an appropriate generalized vorticity and characterized by zero generalized helicity for incompressible fluids. In this article, we investigate these states for compressible plasmas in black hole spacetime geometries using the curved spacetime generalization of the grand generalized vortical formalism. If the magnetic field is axisymmetric and the thermodynamic properties are symmetric about the equatorial plane, the resulting states are characterized by a vanishing skin depth and a complete expulsion of the magnetic field at the equator of the black hole horizon. Moreover, if the thermodynamic properties of the plasma are uniform at the horizon, we find that the magnetic field is completely expelled from the horizon, and the plasma behaves as a perfect superconductor near the horizon. This result is independent of the spin of a black hole, holding even for a (nonrotating) Schwarzschild black hole, and demonstrates that the geometry near black hole horizons can have a significant effect on the electrodynamics of surrounding plasmas.

gr-qc↗

arXiv2018

Dickman approximation in simulation, summations and perpetuities

The generalized Dickman distribution ${\cal D}_θ$ with parameter $θ>0$ is the unique solution to the distributional equality $W=_d W^*$, where \begin{eqnarray} W^*=_d U^{1/θ}(W+1) \qquad (1) \end{eqnarray} with $W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of $W$, and $=_d$ denoting equality in distribution. Members of this family appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between ${\cal D}_θ$ and \begin{eqnarray} W_n= \frac{1}{n} \sum_{i=1}^n Y_k B_k \qquad (2) \end{eqnarray} where $B_1,\ldots,B_n, Y_1, \ldots, Y_n$ are independent with $B_k \sim {\rm Ber}(1/k), E[Y_k]=k, {\rm Var}(Y_k)=σ_k^2$ and provide an application to the minimal directed spanning tree in $\mathbb{R}^2$, and also obtain such bounds when the Bernoulli variables in $(2)$ are replaced by Poissons. We also give simple proofs and provide bounds with optimal rates for the Dickman convergence of the weighted sums, arising in probabilistic number theory, of the form \begin{eqnarray} S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k) \end{eqnarray} where $(p_k)_{k \ge 1}$ is an enumeration of the prime numbers in increasing order and $X_k$ is Geometric with parameter $(1-1/p_k)$, Bernoulli with success probability $1/(1+p_k)$ or Poisson with mean $λ_k$. In addition, we broaden the class of generalized Dickman distributions by studying the fixed points of the transformation \begin{eqnarray*} s(W^*)=_d U^{1/θ}s(W+1) \end{eqnarray*} generalizing $(1)$, that allows the use of non-identity utility functions $s(\cdot)$ in Vervaat perpetuities. We obtain distributional bounds for recursive methods that can be used to simulate from this family.

math.PR↗

arXiv2016

On Strong Embeddings by Stein's Method

Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee's novel use of Stein's method for $\{-1,+1\}$ valued variables to a general class of discrete distributions, and provide $\log n$ rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.

math.PR↗

arXiv2015

Beltrami state in black-hole accretion disk: A magnetofluid approach

Using the magnetofluid unification framework, we show that the accretion disk plasma (embedded in the background geometry of a blackhole) can relax to a class of states known as the Beltrami-Bernoulli (BB) equilibria. Modeling the disk plasma as a Hall MHD system, we find that the space-time curvature can significantly alter the magnetic/velocity decay rate as we move away from the compact object; the velocity profiles in BB states, for example, deviate substantially from the predicted corresponding geodesic velocity profiles. These departures imply a rich interplay of plasma dynamics and general relativity revealed by examining the corresponding Bernoulli condition representing "homogeneity" of total energy. The relaxed states have their origin in the constraints provided by the two helicity invariants of Hall MHD. These helicities conspire to introduce a new oscillatory length scale into the system that is strongly influenced by relativistic and thermal effects.

astro-ph.HE↗