Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.21195

Construction and limit theorems for supCAR fields

Illia Donhauzer·Nikolai Leonenko·Andriy Olenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

The paper introduces a new class of random fields, supCAR fields, which are constructed as superpositions of continuous autoregressive random fields. These supCAR fields possess infinitely divisible marginal distributions. Their second-order properties are characterised by a novel family of covariance functions which can exhibit short- and long-range spatial dependencies. First, the existence of such fields is examined. Then, functional limit theorems for supCAR fields are derived under general assumptions. Four limiting scenarios that depend on the marginals of the underlying autoregressive fields and the specifications of the superposition are identified. Examples of specific supCAR fields, for which the assumptions and results are provided in simple, explicit forms, are presented. The obtained limit theorems can be employed for the statistical inference of supCAR fields.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Illia Donhauzer, Nikolai Leonenko, Andriy Olenko. 2025-05-27. Construction and limit theorems for supCAR fields. https://arxiv.org/abs/2505.21195

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Distribution-uniform strong laws of large numbers

We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on novel distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. We also consider the non-identically distributed case.

math.PR↗
arXiv2026

Malliavin Calculus for rough stochastic differential equations

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

math.PR↗
arXiv2026

Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

math.PR↗