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From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities…

Combinatorics · Mathematics 2007-05-23 A. H. Zemanian

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system. It was proved in [Avigad et al. 2010,…

Numerical Analysis · Computer Science 2010-06-03 Stefano Galatolo , Mathieu Hoyrup , Cristóbal Rojas

Convergence bounds are one of the main tools to obtain information on the performance of a distributed machine learning task, before running the task itself. In this work, we perform a set of experiments to assess to which extent, and in…

Networking and Internet Architecture · Computer Science 2022-12-06 Francesco Malandrino , Carla Fabiana Chiasserini

We study uniform consistency in nonparametric mixture models as well as closely related mixture of regression (also known as mixed regression) models, where the regression functions are allowed to be nonparametric and the error…

Statistics Theory · Mathematics 2022-12-29 Bryon Aragam , Ruiyi Yang

The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary…

Probability · Mathematics 2015-01-22 Sébastien Gouëzel , Frédéric Mathéus , François Maucourant

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…

Geometric Topology · Mathematics 2017-02-21 Jenya Sapir

Confidence sequences are anytime-valid analogues of classical confidence intervals that do not suffer from multiplicity issues under optional continuation of the data collection. As in classical statistics, asymptotic confidence sequences…

Statistics Theory · Mathematics 2025-06-17 Felix Gnettner , Claudia Kirch

This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that (i) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and (ii) there is a known…

Probability · Mathematics 2008-06-19 Andrew B. Nobel , Gusztav Morvai , Sanjeev R. Kulkarni

We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts…

Statistical Mechanics · Physics 2016-01-08 Claude Godreche , Satya N. Majumdar , Gregory Schehr

Continuous time random Walk model has been versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as disordered or porous media. We are studying the continuous limits of Heterogeneous…

Statistical Mechanics · Physics 2020-06-23 Liubov Tupikina

This paper presents convergence acceleration, a method for computing efficiently the limit of numerical sequences as a typical application of streams and higher-order functions.

Numerical Analysis · Computer Science 2014-03-04 Pierre Lescanne

We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices…

Combinatorics · Mathematics 2024-07-29 Mikhail Isaev , Brendan D. McKay , Angus Southwell , Maksim Zhukovskii

Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation.…

Statistical Mechanics · Physics 2024-03-01 Guoxing Lin , Shaokun Zheng

We derive a general upper bound for the number of incidences with $k$-dimensional varieties in ${\mathbb R}^d$. The leading term of this new bound generalizes previous bounds for the special cases of $k=1, k=d-1,$ and $k= d/2$, to every…

Combinatorics · Mathematics 2018-09-13 Thao Do , Adam Sheffer

We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We…

Probability · Mathematics 2018-11-27 Amir Dembo , Ryoki Fukushima , Naoki Kubota

Often in the analysis of first-order methods, assuming the existence of a quadratic growth bound (a generalization of strong convexity) facilitates much stronger convergence analysis. Hence the analysis is done twice, once for the general…

Optimization and Control · Mathematics 2019-05-16 Benjamin Grimmer

In this paper, we develop a general machinery for finding explicit uniform probability and moment bounds on sub-additive positive functionals of random processes. Using the developed general technique, we derive uniform bounds on the…

Probability · Mathematics 2012-02-09 Alexander Goldenshluger , Oleg Lepski

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and…

Dynamical Systems · Mathematics 2011-02-09 Qing Chu , Nikos Frantzikinakis

The vast majority of the literature on learning dynamical systems or stochastic processes from time series has focused on stable or ergodic systems, for both Bayesian and frequentist inference procedures. However, most real-world systems…

Machine Learning · Statistics 2025-07-02 Zachary P Adams , Sayan Mukherjee

This paper proposes novel tests for the absence of jumps in a univariate semimartingale and for the absence of common jumps in a bivariate semimartingale. Our methods rely on ratio statistics of power variations based on irregular…

Statistics Theory · Mathematics 2017-12-21 Ole Martin , Mathias Vetter