Asymptotic Analysis of Random Walks: Light-Tailed Distributions

Боровков, А. А.; Ulyanov, Vladimir V.; Zhitlukhin, Mikhail

doi:10.1017/9781139871303

Cited by 47 publications

(52 citation statements)

References 126 publications

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“…If stronger assumptions are made on , such as regular variation, then corresponding lower bounds are known for certain sets A , but it remains unclear whether or not our abstract approach can recover such lower bounds. Refer to [13], [29], and [41] for detailed overviews of such results, as well as the more recent [17], [49], and references therein. Indeed, precise asymptotics require detailed assumptions on the shape of the tails of , and this is especially true in multivariate and infinite-dimensional contexts.…”

Section: Introductionmentioning

confidence: 99%

A non-exponential extension of Sanov’s theorem via convex duality

Lacker

2020

Adv. Appl. Probab.

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This work is devoted to a vast extension of Sanov's theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of i.i.d. samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include uniform large deviation bounds, variational problems involving optimal transport costs, and constrained super-hedging problems, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis-Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

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Section: Introductionmentioning

confidence: 99%

A non-exponential extension of Sanov’s theorem via convex duality

Lacker

2020

Adv. Appl. Probab.

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“…uniformly in u/t ∈ [0, 1]. Broad sufficient conditions for the validity of (4) can be found, e.g., in Theorem 9.3.1 of [Borovkov, Borovkov, 2008].…”

Section: Model Description and Main Resultsmentioning

confidence: 99%

Isotropic Multidimensional Catalytic Branching Random Walk with Regularly Varying Tails

Bulinskaya¹

2019

CRM

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The study completes a series of the author's works devoted to the spread of particles population in supercritical catalytic branching random walk (CBRW) on a multidimensional lattice. The CBRW model describes the evolution of a system of particles combining their random movement with branching (reproduction and death) which only occurs at fixed points of the lattice. The set of such catalytic points is assumed to be finite and arbitrary. In the supercritical regime the size of population, initiated by a parent particle, increases exponentially with positive probability. The rate of the spread depends essentially on the distribution tails of the random walk jump. If the jump distribution has "light tails", the "population front", formed by the particles most distant from the origin, moves linearly in time and the limiting shape of the front is a convex surface. When the random walk jump has independent coordinates with a semiexponential distribution, the population spreads with a power rate in time and the limiting shape of the front is a star-shape nonconvex surface. So far, for regularly varying tails ("heavy" tails), we have considered the problem of scaled front propagation assuming independence of components of the random walk jump. Now, without this hypothesis, we examine an "isotropic" case, when the rate of decay of the jumps distribution in different directions is given by the same regularly varying function. We specify the probability that, for time going to infinity, the limiting random set formed by appropriately scaled positions of population particles belongs to a set B containing the origin with its neighborhood, in R d . In contrast to the previous results, the random cloud of particles with normalized positions in the time limit will not concentrate on coordinate axes with probability one.

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“…That is why different approaches have been developed, among which large deviation theorems (see e.g. Petrov (1975) and Borovkov (2020) for light tails and Mikosch and Nagaev (1998), Foss et al (2013), and Lehtomaa (2017) for heavy tails, and references therein), extreme value theorems (EVT) focusing on the tail only (see e.g. Embrechts et al (1997), de Haan and Ferreira (2006), and Resnick (2007)), and hybrid distributions combining (asymptotic) distributions for both the main and extreme behaviors when considering independent random variables (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence

Kratz¹,

Prokopenko²

2021

Preprint

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We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'normex' approach from a univariate to a multivariate framework. We propose two possible multinormex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles.

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Asymptotic Analysis of Random Walks: Light-Tailed Distributions

Cited by 47 publications

References 126 publications

A non-exponential extension of Sanov’s theorem via convex duality

A non-exponential extension of Sanov’s theorem via convex duality

Isotropic Multidimensional Catalytic Branching Random Walk with Regularly Varying Tails

Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence

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