arXiv subjects

Daniel Raban

Publications and source records attributed to Daniel Raban.

5 recordsLinked to original sources

arXiv2026

Self-Balancing Sequential Sampling: Fast Convergence with Controlled Predictability

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

math.PR↗

arXiv2025

Coupling Theory, Optimal Transport, and Strassen's Theorem for Eventual Domination

Many results in probability (most famously, Strassen's theorem on stochastic domination), characterize some relationship between probability distributions in terms of the existence of a particular structured coupling between them. Optimal transport, and in particular Kantorovich duality, provides a common framework for these results. We use this perspective to study "eventual domination", a class of orders arising naturally in stochastic processes (including in the analysis of branching random walks, Ising models, and diffusions), which does not satisfy the topological conditions required by standard optimal transport theory. More generally, we study the connection between distributional relations and their coupling counterparts for topologically irregular preorders (e.g. equivalence relations, partial orders). To this end, we show that Strassen's theorem "nearly holds" in this topologically irregular setting but that the full theorem admits counterexamples, including for eventual domination. The core of the proof is a novel technical result in optimal transport, which shows that the Kantorovich dual problem is well-behaved for cost functions which can be written as a non-increasing limit of lower semi-continuous functions.

math.PR↗

arXiv2025

Laws of Large Numbers for Information Resolution

Laws of large numbers establish asymptotic guarantees for recovering features of a probability distribution using independent samples. We introduce a framework for proving analogous results for recovery of the $\sigma$-field of a probability space, interpreted as information resolution--the granularity of measurable events given by comparison to our samples. Our main results show that, under iid sampling, the Borel $\sigma$-field in $\mathbb R^d$ and in more general metric spaces can be recovered in the strongest possible mode of convergence. We also derive finite-sample $L^1$ bounds for uniform convergence of $\sigma$-fields on $[0,1]^d$. We illustrate the use of our framework with two applications: constructing randomized solutions to the Skorokhod embedding problem, and analyzing the loss of variants of random forests for regression.

math.PR↗

arXiv2024

How should we aggregate ratings? Accounting for personal rating scales via Wasserstein barycenters

A common method of comparing items is to collect numerical ratings on a linear scale and compare the average rating for each item. However, averaging ratings does not account for people rating according to differing personal rating scales. With this in mind, we investigate the problem of calculating aggregate numerical ratings from individual numerical ratings and propose a new, non-parametric model for the problem. We show that, with minimal modeling assumptions, the standard average is inconsistent for estimating the quality of items. Analyzing the problem of heterogeneous personal rating scales from the perspective of optimal transport, we derive an alternative rating estimator, which we show is asymptotically consistent almost surely and in L^p for estimating quality, with an optimal rate of convergence. Further, we generalize Kendall's W, a non-parametric coefficient of preference concordance between raters, from the special case of rankings to the more general case of arbitrary numerical ratings. Along the way, we prove Glivenko--Cantelli-type theorems for uniform convergence of the cumulative distribution functions and quantile functions for Wasserstein-2 barycenters on [0,1].

math.ST↗

arXiv2019

Rotatable random sequences in local fields

An infinite sequence of real random variables $(ξ_1, ξ_2, \dots)$ is said to be rotatable if every finite subsequence $(ξ_1, \dots, ξ_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(ξ_1, ξ_2, \dots)$ is rotatable if and only if $ξ_j = τη_j$ for all $j$, where $(η_1, η_2, \dots)$ is a sequence of independent standard Gaussian random variables and $τ$ is an independent nonnegative random variable. Freedman's theorem is equivalent to a classical result of Schoenberg which says that a continuous function $ϕ: \mathbb{R}_+ \to \mathbb{C}$ with $ϕ(0) = 1$ is completely monotone if and only if $ϕ_n: \mathbb{R}^n \to \mathbb{R}$ given by $ϕ_n(x_1, \ldots, x_n) = ϕ(x_1^2 + \cdots + x_n^2)$ is nonnegative definite for all $n \in \mathbb{N}$. We establish the analogue of Freedman's theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenberg's result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincaré, and Borel which says that if $(ζ_1, \ldots, ζ_n)$ is uniformly distributed on the sphere of radius $\sqrt{n}$ in $\mathbb{R}^n$, then, for fixed $k \in \mathbb{N}$, the distribution of $(ζ_1, \ldots, ζ_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n \to \infty$.

math.PR↗