Related papers: Implicit renewal theory in the arithmetic case

Maxwell's equations can be solved numerically in space manifold and the time by discrete exterior calculus as a kind of lattice gauge theory.Since the stable conditions of this method is very severe restriction, we combine the implicit…

Stochastic volatility processes with heavy-tailed innovations are a well-known model for financial time series. In these models, the extremes of the log returns are mainly driven by the extremes of the i.i.d. innovation sequence which leads…

We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$,…

We establish a new integral equation for the probability density of the exponential functional of a L\'evy process and provide a three-term (Wiener-Hopf type) factorisation of its law. We explain how these results complement the techniques…

We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cram\'er condition the stationary distribution has a power law tail. We…

We consider the problem of computing the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=\sum_{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\ge -1$, are nonnegative square matrices such that $\sum_{i=-1}^\infty A_i$ is…

We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest. We obtain first and second order…

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove the analogue of de Finetti's theorem for this type of processes. In addition, since the vacuum state on the $q$-deformed $C^*$-algebra is…

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…

Extreme events and the heavy tail distributions driven by them are ubiquitous in various scientific, engineering and financial research. They are typically associated with stochastic instability caused by hidden unresolved processes.…

Estimation of tail quantities, such as expected shortfall or Value at Risk, is a difficult problem. We show how the theory of nonlinear expectations, in particular the Data-robust expectation introduced in [5], can assist in the…

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation…

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to…

We propose a family of models that enable predictive estimation of time-varying extreme event probabilities in heavy-tailed and nonlinearly dependent time series. The models are a white noise process with conditionally log-Laplace…

Multivariate regular variation plays a role assessing tail risk in diverse applications such as finance, telecommunications, insurance and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to…

For a given random sequence $(C,T_{1},T_{2},\ldots)$ with nonzero $C$ and a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform $\mathcal{S}$ maps the law of a real random variable $X$ to the law of $\sum_{k\ge…

Let $X$ be a random vector in $\rd$ with a regularly varying tail. We consider two transformations $\|X\|f(\frac{X}{\|X\|})$, $f: \sd\to\sd$, and $Xf(\frac{X}{\|X\|})$, $f: \sd\to \mathbb{R}_+$. Some sufficient conditions for preserving the…

In dimension $d\geq3$, we present a general assumption under which the renewal theorem established by Spitzer for i.i.d. sequences of centered nonlattice r.v. holds true. Next we appeal to an operator-type procedure to investigate the…

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…

Stochastic ordering of distributions of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual…