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Nikola Sandrić

Publications and source records attributed to Nikola Sandrić.

At least 19 recordsLinked to original sources
arXiv2026

Fluctuations for diameter and perimeter of convex hulls of multiple random walks

We study the diameter and perimeter of the convex hull generated by finitely many independent planar random walks whose increments have finite second moments. The large-time fluctuations are governed by the geometry of the polygon formed by the drift vectors. We develop an $L^2$-approximation framework, based on Wald-type maximal central limit theorems, which reduces the asymptotic analysis of the hull to a finite collection of endpoint, maximal-projection, and Brownian support-function terms. For the diameter, we obtain general max-type limit theorems, Gaussian in the case of a unique extremal diametrical pair and typically non-Gaussian when several extremal pairs compete. For the perimeter, we prove a general distributional limit: non-zero extremal drifts contribute maxima of Gaussian projections, while zero-drift extremal walks contribute Brownian support-function terms. The results recover the previously known Gaussian regimes (the case of one or two walks) and identify the non-Gaussian limits in the degenerate and boundary cases left open (even for two walks). We also give $L^2$ approximations of the convex hull by simpler random sets, under Hausdorff and $\ell_1$ metrics on compact convex sets. Our proofs work under the optimal finite second moment assumption.

math.PR↗
arXiv2026

On the convex hull of a planar Brownian bridge with a random Gaussian endpoint

We consider a one-parameter family of isotropic planar Gaussian processes \[ X_σ(t) =B_t+σt Z,\qquad 0\le t\le 1,\quad 0\le σ\le 1, \] where $B$ is a standard ($0$-to-$0$) planar Brownian bridge on $[0,1]$, and $Z\sim \mathrm N(0,I)$ is a standard Gaussian random vector independent of $B$. The family interpolates between standard planar Brownian bridge ($σ=0$) and standard planar Brownian motion ($σ=1$). As the main result of the paper we compute the expected perimeter and area of the convex hull of the random set $\left\{X_σ(t) \colon 0\le t\le 1\right\}$ as closed formulas in terms of $σ$, and recover the classical Brownian bridge and Brownian motion values at $σ=0$ and $σ=1$. We also consider the convex hull spanned by multiple independent processes of this type and the possibilities for closed formulas in special cases. The key observation in our argument is that the isotropy property reduces the expected perimeter and area to one-dimensional quantities through the support function and Cauchy's formulas.

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

On the convex hull of two planar random walks

In this paper, we study the limiting behavior of the perimeter and diameter functionals of the convex hull spanned by the first $n$ steps of two planar random walks. As the main results, we obtain the strong law of large numbers and the central limit theorem for the perimeter and diameter of these random sets.

math.PR↗
arXiv2025

Rademacher learning rates for iterated random functions

Most existing literature on supervised machine learning assumes that the training dataset is drawn from an i.i.d. sample. However, many real-world problems exhibit temporal dependence and strong correlations between the marginal distributions of the data-generating process, suggesting that the i.i.d. assumption is often unrealistic. In such cases, models naturally include time-series processes with mixing properties, as well as irreducible and aperiodic ergodic Markov chains. Moreover, the learning rates typically obtained in these settings are independent of the data distribution, which can lead to restrictive choices of hypothesis classes and suboptimal sample complexities for the learning algorithm. In this article, we consider the case where the training dataset is generated by an iterated random function (i.e., an iteratively defined time-homogeneous Markov chain) that is not necessarily irreducible or aperiodic. Under the assumption that the governing function is contractive with respect to its first argument and subject to certain regularity conditions on the hypothesis class, we first establish a uniform convergence result for the corresponding sample error. We then demonstrate the learnability of the approximate empirical risk minimization algorithm and derive its learning rate bound. Both rates are data-distribution dependent, expressed in terms of the Rademacher complexities of the underlying hypothesis class, allowing them to more accurately reflect the properties of the data-generating distribution.

stat.ML↗
arXiv2025

A note on the uniform ergodicity of diffusion processes

In this note, we discuss the uniform ergodicity of a diffusion process given by an Itô stochastic differential equation. We present an integral condition in terms of the drift and diffusion coefficients that ensures the uniform ergodicity of the corresponding transition kernel with respect to the total variation distance. Applications of the obtained results to a class of subordinate diffusion processes are also presented.

math.PR↗
arXiv2024

Iterated-logarithm laws for convex hulls of random walks with drift

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.

math.PR↗
arXiv2023

Periodic homogenization of a class of weakly coupled systems of linear PDEs

In this article, basing upon probabilistic methods, we discuss periodic homogenization of a class of weakly coupled systems of linear elliptic and parabolic partial differential equations. Under the assumption that the systems have rapidly periodically oscillating coefficients, we first prove that the appropriately centered and scaled continuous component of the associated regime switching diffusion process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the systems. The homogenization results then follow by employing probabilistic representation of the solutions to the systems and the continuous mapping theorem. The presented results generalize the well-known results related to periodic homogenization of the classical elliptic boundary-value problem and the classical parabolic initial-value problem for a single equation.

math.PR↗
arXiv2023

Learning from non-irreducible Markov chains

Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong correlation between the marginals of the data-generating process, suggesting that the i.i.d. assumption is not always justified. This problem has been already considered in the context of Markov chains satisfying the Doeblin condition. This condition, among other things, implies that the chain is not singular in its behavior, i.e. it is irreducible. In this article, we focus on the case when the training data set is drawn from a not necessarily irreducible Markov chain. Under the assumption that the chain is uniformly ergodic with respect to the $\mathrm{L}^1$-Wasserstein distance, and certain regularity assumptions on the hypothesis class and the state space of the chain, we first obtain a uniform convergence result for the corresponding sample error, and then we conclude learnability of the approximate sample error minimization algorithm and find its generalization bounds. At the end, a relative uniform convergence result for the sample error is also discussed.

math.ST↗
arXiv2022

On subgeometric ergodicity of regime-switching diffusion processes

In this article, we discuss subgeometric ergodicity of a class of regime-switching diffusion processes. We derive conditions on the drift and diffusion coefficients, and the switching mechanism which result in subgeometric ergodicity of the corresponding semigroup with respect to the total variation distance as well as a class of Wasserstein distances. At the end, subgeometric ergodicity of certain classes of regime-switching Markov processes with jumps is also discussed.

math.PR↗
arXiv2022

Subexponential upper and lower bounds in Wasserstein distance for Markov processes

In this article, relying on Foster-Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the $\mathrm{L}^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov Lévy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the $\mathrm{L}^p$-Wasserstein distance for a class of Itô processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein-Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds.

math.PR↗
arXiv2022

Convex hulls of stable random walks

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.

math.PR↗
arXiv2021

Capacity of the range of random walks on groups

In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem.

math.PR↗
arXiv2021

CLT for the capacity of the range of stable random walks

In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $α$-stable random walks with the index satisfying $d > 5α/2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.

math.PR↗
arXiv2020

Limit theorems for a stable sausage

In this article, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $α$-stable process. As the main results, we establish a functional central limit theorem (in the case when $d/α>3 /2$) with a standard one-dimensional Brownian motion in the limit, and Khintchine's and Chung's laws of the iterated logarithm (in the case when $d/α>9 /5$).

math.PR↗
arXiv2020

Functional CLT for the range of stable random walks

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $α$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5α/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3α/ 2$.

math.PR↗