Related papers: Uniform convergence for complex $[\mathbf{0,1}]$-m…
In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure. This makes it much harder to describe extremals and to attack such problems. Many of these problems are related to the…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise,…
We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities…
We provide finite-sample distribution approximations, that are uniform in the parameter, for inference in linear mixed models. Focus is on variances and covariances of random effects in cases where existing theory fails because their…
We obtain an uniform tail estimates for natural normed sums of independent random variables (r.v.) with regular varying tails of distributions. We give also many examples on order to show the exactness of offered estimates and discuss some…
We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T\colon \RR\to \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we…
We establish a practical and easy-to-implement sequential stopping rule for the martingale central limit theorem, focusing on Monte Carlo methods for estimating the mean of a non-iid sequence of martingale difference type. Starting with an…
We generalise the randomness test definitions in the literature for both the Martin-L\"of and Schnorr randomness of a series of binary outcomes, in order to allow for interval-valued rather than merely precise forecasts for these outcomes,…
We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set.…
In Monte-Carlo methods the Markov processes used to sample a given target distribution usually satisfy detailed balance, i.e. they are time-reversible. However, relatively recent results have demonstrated that appropriate reversible and…
In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu$ and $\nu$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with…
Universal properties of turbulence have been associated traditionally with very high Reynolds numbers, but recent work has shown that the onset of the power-laws in derivative statistics occurs at modest microscale Reynolds numbers of the…
Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry,…
We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting…
We extend the spectral method for proving limit theorems to random non-uniformly expanding dynamical systems. This yields the CLT and moderate deviations principles (MDP). We show that as the amount of non-uniformity decreases the CLT rates…
We consider uniform moment convergence of lag-window spectral density estimates for univariate and multivariate stationary processes. Optimal rates of convergence are obtained under mild and easily verifiable conditions. Our theory…
Some classes of increment martingales, and the corresponding localized classes, are studied. An increment martingale is indexed by the real line and its increment processes are martingales. We focus primarily on the behavior as time goes to…
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration…
We extend the approach of Walker (2003, 2004) to the case of misspecified models. A sufficient condition for establishing rates of convergence is given based on a key identity involving martingales, which does not require construction of…
Nerman's martingale plays a central role in the law of large numbers for both, single- and multi-type, supercritical general branching processes. There are further, complex-valued Nerman-type martingales in the single-type process that…