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Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic…
Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account on various martingale-based methods for over- and under-approximating…
We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven when we…
In this note we prove estimates for the average cost in the quadratic optimal transport problem on the two-dimensional flat torus which are optimal up to a double logarithm. We also prove sharp estimates on the displacement. This is based…
Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered…
We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and for stationary sequences the…
A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random…
We establish a rather sharp two-side estimate for the tail probability of the derivative martingale limit in a branching random walk throughout the entire subcritical regime, confirming a conjecture by Lacoin, Rhodes, and Vargas (\emph{Duke…
We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random variables by Maxwell and Woodroofe. Our approach is based on new results for…
In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman $N$-coalescent back from time $t$ consider the associated processes of total tree length as $t$ increases. We show that the…
Given a finite honest time, we first show that the associated Az\'ema optional supermartingale can be expressed as the drawdown and the relative drawdown of some local optional supermartingales with continuous running supremum. The relative…
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…
In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to 0. The function of the $i$th increment may depend on the current time, and also…
This paper studies the chance constrained fractional programming with a random benchmark. We assume that the random variables on the numerator follow the Gaussian distribution, and the random variables on the denominator and the benchmark…
In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of…
Recently Avis and Jordan have demonstrated the efficiency of a simple technique called budgeting for the parallelization of a number of tree search algorithms. The idea is to limit the amount of work that a processor performs before it…
For a branching random walk that drifts to infinity, consider its Malthusian martingale, i.e.~the additive martingale with parameter $\theta$ being the smallest root of the characteristic equation. When particles are killed below the…
Recently, the contributions of chiral logarithms predicted by quenched chiral perturbation theory have been extracted from lattice calculations of hadron masses. We argue that a detailed comparison of random matrix theory and lattice…
We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or…
Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of…