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Aleksandar Mijatović

Publications and source records attributed to Aleksandar Mijatović.

At least 19 recordsLinked to original sources
arXiv2026

Hypocoercivity of Tempered Bouncy Particle Samplers for Heavy-Tailed Targets

We introduce the Tempered Bouncy Particle Sampler, a state-dependently tempered generalisation of the Bouncy Particle Sampler. Adapting the hypocoercivity framework of Dolbeault, Mouhot and Schmeiser (2010), we establish explicit, non-asymptotic exponential convergence bounds for a broad class of heavy-tailed targets that need not be log-concave or radially symmetric, including distributions with polynomial tails. The convergence rate is linked to the spectral gap of a tempered Langevin diffusion with the same invariant distribution, making a weighted Poincaré inequality the key assumption. We give admissible tempering choices for polynomial and stretched-exponential tails and derive mixing-time bounds with polynomial dependence on the dimension.

math.PR↗
arXiv2026

Infinite-variation Lévy processes are either strongly eroded or abrupt

We prove that every real Lévy process of infinite variation is either abrupt or strongly eroded, resolving the conjectured geometric dichotomy on the smoothness of the convex hull of the graph of a Lévy path [Ann. Inst. H. Poincaré Probab. Stat. 60(4) (2024), 2352-2381]. We give counterexamples to the stronger point-hitting conjecture of Vigon. As applications, we classify convex-hull smoothness, Lipschitz-minorant contact sets and Burgers shock geometry.

math.PR↗
arXiv2026

Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion

Motivated by semi-Markov models of anomalous diffusion, we study obliquely reflected diffusions in generalized parabolic domains under independent inverse-subordinator time changes. We focus on infinite-mean subordinators, whose inverse clocks can slow the superdiffusive growth induced by reflection. For domains with cross-sectional radius asymptotic to $a_\infty x^β$, where $-1<β<1$, we strengthen the almost-sure growth laws of [MMW23] to obtain precise second-moment asymptotics when the subordinator's Laplace exponent is regularly varying at zero, and matching bounds more generally. Inverse $α$-stable clocks yield mean-square growth of order $t^{2α/(1+β)}$, giving subdiffusive, diffusive, or superdiffusive regimes. For $β<-1$, explosion persists almost surely, with finite mean explosion time if and only if the subordinator has finite mean. Supporting results establish strong existence, pathwise uniqueness, and a time-change representation, also allowing dependent clocks.

math.PR↗
arXiv2026

Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains

We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy--Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.

math.PR↗
arXiv2026

Superdiffusive limits for stochastic kinetics driven by self-similar drifts

We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic dynamics, in which the stochastic drift is driven by an exogenous self-similar noise, and also includes endogenous volatility which is permitted to have arbitrary dependence with the exogenous noise. We identify the superdiffusive scaling exponent for the model, and prove strong and weak convergence results on the corresponding scale. Our framework admits self-similar noise that is either a Bessel process, or, more generally, a self-similar continuous-state branching process with immigration, as well as more general processes satisfying certain asymptotic conditions.

math.PR↗
arXiv2026

Diffeomorphic Markov Chain Monte Carlo: fast mixing for heavy-tailed distributions

We introduce a new class of uniformly ergodic MCMC algorithms, termed Diffeomorphic Contraction Sampler (DCS), and provide fast non-asymptotic mixing guarantees for DCS targeting distributions on $\R^d$ with arbitrarily heavy polynomial tails. DCS provides a solution to a well-known problem for MCMC samplers, which typically struggle with the combination of unbounded high-dimensional state space and vanishing gradients. The DCS pulls back a target on $\R^d$ onto a Euclidean ball $B(R)\subset\R^d$ and then samples from the transformed density on the convex set $B(R)$ via algorithms such as the Ball Walk, Hit-and-Run and others. A radial diffeomorphic contraction is chosen so that the pull-back density on $B(R)$ is bounded, implying uniform ergodicity for \textit{all} targets with a finite polynomial moment. Non-asymptotic bounds for DCS require stronger assumptions such as log-concavity of the pull-back density. In practice, this is achieved approximately by a preconditioned automorphism of the ball $B(R)$, tuned via Variational Inference. Numerical simulation tests demonstrate that the DCS outperforms significantly the No-U-Turns sampler on multi-dimensional heavy-tailed targets arising as real-world posteriors in PosteriorDB benchmark. DCS also numerically outperforms in high-dimensional examples recently developed spherical projection samplers for heavy-tailed target distributions.

stat.CO↗
arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗
arXiv2025

Central Limit Theorem for ergodic averages of Markov chains \& the comparison of sampling algorithms for heavy-tailed distributions

Establishing central limit theorems (CLTs) for ergodic averages of Markov chains is a fundamental problem in probability and its applications. Since the seminal work~\cite{MR834478}, a vast literature has emerged on the sufficient conditions for such CLTs. To counterbalance this, the present paper provides verifiable necessary conditions for CLTs of ergodic averages of Markov chains on general state spaces. Our theory is based on drift conditions, which also yield lower bounds on the rates of convergence to stationarity in various metrics. The validity of the ergodic CLT is of particular importance for sampling algorithms, where it underpins the error analysis of estimators in Bayesian statistics and machine learning. Although heavy-tailed sampling is of central importance in applications, the characterisation of the CLT and the convergence rates are theoretically poorly understood for almost all practically-used Markov chain Monte Carlo (MCMC) algorithms. In this setting our results provide sharp conditions on the validity of the ergodic CLT and establish convergence rates for large families of MCMC sampling algorithms for heavy-tailed targets. Our study includes a rather complete analyses for random walk Metropolis samplers (with finite- and infinite-variance proposals), Metropolis-adjusted and unadjusted Langevin algorithms and the stereographic projection sampler (as well as the independence sampler). By providing these sharp results via our practical drift conditions, our theory offers significant insights into the problems of algorithm selection and comparison for sampling heavy-tailed distributions (see short YouTube presentations~\cite{YouTube_talk} describing our \href{https://youtu.be/m2y7U4cEqy4}{\underline{theory}} and \href{https://youtu.be/w8I_oOweuko}{\underline{applications}}).

math.PR↗
arXiv2025

Stability of storage processes with general release rates

This paper quantifies the ergodicity and the rate of decay of the tail of the stationary distribution for a broad class of storage models, encompassing constant, linear, and power-type release rates with both finite and infinite activity input process. Our results are expressed in terms of the asymptotics of the release rate, the tail-decay rate of the Lévy measure of the input process and its (possibly infinite) first moment. Our framework unifies and significantly extends classical results on the stability of storage models. Under certain regularity assumptions, we also provide upper bounds on the stability in the Wasserstein distance.

math.PR↗
arXiv2025

Multivariate CLT for Lévy processes: convergence rates without moment assumptions

We prove that the norm of a $d$-dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\infty)$. We further prove that under the standard $\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the Lévy process has a finite $(2+\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for Lévy processes cannot hold without finiteness of $(2+δ)$-moments for some $δ>0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.

math.PR↗
arXiv2025

Limit Theorems for Stochastic Gradient Descent with Infinite Variance

Stochastic gradient descent is a classic algorithm that has gained great popularity especially in the last decades as the most common approach for training models in machine learning. While the algorithm has been well-studied when stochastic gradients are assumed to have a finite variance, there is significantly less research addressing its theoretical properties in the case of infinite variance gradients. In this paper, we establish the asymptotic behavior of stochastic gradient descent in the context of infinite variance stochastic gradients, assuming that the stochastic gradient is regular varying with index $α\in(1,2)$. The closest result in this context was established in 1969 , in the one-dimensional case and assuming that stochastic gradients belong to a more restrictive class of distributions. We extend it to the multidimensional case, covering a broader class of infinite variance distributions. As we show, the asymptotic distribution of the stochastic gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-Uhlenbeck process driven by an appropriate stable Lévy process. Additionally, we explore the applications of these results in linear regression and logistic regression models.

stat.ML↗
arXiv2025

Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat

Denoising diffusion probabilistic models (DDPMs) represent a recent advance in generative modelling that has delivered state-of-the-art results across many domains of applications. Despite their success, a rigorous theoretical understanding of the error within DDPMs, particularly the non-asymptotic bounds required for the comparison of their efficiency, remain scarce. Making minimal assumptions on the initial data distribution, allowing for example the manifold hypothesis, this paper presents explicit non-asymptotic bounds on the forward diffusion error in total variation (TV), expressed as a function of the terminal time $T$. We parametrise multi-modal data distributions in terms of the distance $R$ to their furthest modes and consider forward diffusions with additive and multiplicative noise. Our analysis rigorously proves that, under mild assumptions, the canonical choice of the Ornstein-Uhlenbeck (OU) process cannot be significantly improved in terms of reducing the terminal time $T$ as a function of $R$ and error tolerance $\varepsilon>0$. Motivated by data distributions arising in generative modelling, we also establish a cut-off like phenomenon (as $R\to\infty$) for the convergence to its invariant measure in TV of an OU process, initialized at a multi-modal distribution with maximal mode distance $R$.

math.ST↗
arXiv2025

Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction

We develop two novel couplings between general pure-jump Lévy processes in $\R^d$ and apply them to obtain upper bounds on the rate of convergence in an appropriate Wasserstein distance on the path space for a wide class of Lévy processes attracted to a multidimensional stable process in the small-time regime. We also establish general lower bounds based on certain universal properties of slowly varying functions and the relationship between the Wasserstein and Toscani--Fourier distances of the marginals. Our upper and lower bounds typically have matching rates. In particular, the rate of convergence is polynomial for the domain of normal attraction and slower than a slowly varying function for the domain of non-normal attraction.

math.PR↗
arXiv2025

Critical branching processes with immigration: scaling limits of local extinction sets

We establish the joint scaling limit of a critical Bienaymé-Galton-Watson process with immigration (BGWI) and its (counting) local time at zero to the corresponding self-similar continuous-state branching process with immigration (CBI) and its (Markovian) local time at zero for balanced offspring and immigration laws in stable domains of attraction. Using a general framework for invariance principles of local times~\cite{MR4463082}, the problem reduces to the analysis of the structure of excursions from zero and positive levels, together with the weak convergence of the hitting times of points of the BGWI to those of the CBI. A key step in the proof of our main limit theorem is a novel Yaglom limit for the law at time $t$ of an excursion with lifetime exceeding $t$ of a scaled infinite-variance critical BGWI. Our main result implies a joint septuple scaling limit of BGWI $Z_1$, its local time at $0$, the random walks $X_1$ and $Y_1$ associated to the reproduction and immigration mechanisms, respectively, the counting local time at $0$ of $X_1$, an additive functional of $Z_1$ and $X_1$ evaluated at this functional. In the septuple limit, four different scaling sequences are identified and given explicitly in terms of the offspring generating function (modulo asymptotic inversion), the local extinction probabilities of the BGWI and the tails of return times to zero of $X_1$.

math.PR↗
arXiv2024

Central limit theorem for superdiffusive reflected Brownian motion

We study the second-order asymptotics around the superdiffusive strong law~\cite{MMW} of a multidimensional driftless diffusion with oblique reflection from the boundary in a generalised parabolic domain. In the unbounded direction we prove the limit is Gaussian with the usual diffusive scaling, while in the appropriately scaled cross-sectional slice we establish convergence to the invariant law of a reflecting diffusion in a unit ball. Using the separation of time scales, we also show asymptotic independence between these two components. The parameters of the limit laws are explicit in the growth rate of the boundary and the asymptotic diffusion matrix and reflection vector field. A phase transition occurs when the domain becomes too narrow, in which case we prove that the central limit theorem for the unbounded component fails.

math.PR↗
arXiv2024

Modeling of Measurement Error in Financial Returns Data

In this paper we consider the modeling of measurement error for fund returns data. In particular, given access to a time-series of discretely observed log-returns and the associated maximum over the observation period, we develop a stochastic model which models the true log-returns and maximum via a Lévy process and the data as a measurement error there-of. The main technical difficulty of trying to infer this model, for instance Bayesian parameter estimation, is that the joint transition density of the return and maximum is seldom known, nor can it be simulated exactly. Based upon the novel stick breaking representation of [12] we provide an approximation of the model. We develop a Markov chain Monte Carlo (MCMC) algorithm to sample from the Bayesian posterior of the approximated posterior and then extend this to a multilevel MCMC method which can reduce the computational cost to approximate posterior expectations, relative to ordinary MCMC. We implement our methodology on several applications including for real data.

stat.CO↗
arXiv2024

Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function

We construct a fast exact algorithm for the simulation of the first-passage time, jointly with the undershoot and overshoot, of a tempered stable subordinator over an arbitrary non-increasing absolutely continuous function. We prove that the running time of our algorithm has finite exponential moments and provide bounds on its expected running time with explicit dependence on the characteristics of the process and the initial value of the function. The expected running time grows at most cubically in the stability parameter (as it approaches either $0$ or $1$) and is linear in the tempering parameter and the initial value of the function. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds. We provide numerical examples to illustrate the performance of our algorithm in Monte Carlo estimation.

math.PR↗