We study the random variables (r.v.) with values in the so-called mixed (anisotropic) Lebesgue-Riesz spaces: formulate the sufficient conditions for belonging of the r.v. to these spaces, estimate the tail of norms distribution, especially deduce the exponential decreasing tails of them, etc. We obtain as a consequence the estimations of the norms of random integral operators acting between these spaces.
We deduce sufficient conditions for the Central Limit Theorem (CLT) in the Lebesgue-Riesz space L(p) defined on some measure space for the sequence of centered random variables satisfying the strong mixing (Rosenblatt) condition. We investigate the asymptotical as well as non-asymptotical approach.
We study the Central Limit Theorem (CLT) in the so-called mixed (anisotropic) Lebesgue-Riesz spaces and tail behavior of normed sums of centered random independent variables (vectors) with values in these spaces.
We bring in this short report (counter -) examples on order to show a difference between tails behavior and Grand Lebesgue Spaces norms for some classes of random variables.
We obtain the quite exact exponential bounds for tails of distributions of sums of Banach space valued random variables uniformly over the number of summands under natural for the Law of Iterated Logarithm (LIL) norming. We study especially the case of the so-called mixed (anisotropic) Lebesgue-Riesz spaces, on the other words, Bochner's spaces, for instance, continuous-Lebesgue spaces, which appear for example in the investigation of non-linear Partial Differential Equations of evolutionary type. We give also some examples in order to show the exactness of our estimates.
We study the Central Limit Theorem (CLT) in the so-called hybrid Lebesgue-continuous spaces and tail behavior of normed sums of centered random independent variables (vectors) with values in these spaces.
We offer in this paper the non-asymptotical pairwise bilateral exact up to multiplicative constants interrelations between the tail behavior, moments (Grand Lebesgue Spaces) norm and Orlicz's norm for random variables (r.v.), which does not satisfy in general case the Cramer's condition.
For independent $X$ and $Y$ in the inequality $P(X\leq Y+μ)$, we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).
We present in this paper a new sufficient condition for the so-called Prokhorov-Skorokhod continuity of random processes. Our conditions will be formulated in the terms of metric entropy generated by three-dimensional distribution of the considered random process (r.p.) in the parametric set, have a convenient and closed form, and generalize some previous results. We study also the conditions for weak compactness of the sequence of random processes in this space and as a consequence the Central Limit Theorem. Our consideration based on the theory of Prokhorov-Skorokhod spaces of random process…
We derive the tail inequalities between two random variables starting from inequalities between its moment, or more generally between its Lebesgue-Riesz norms, which holds true on certain sets of parameters. We consider some applications into the martingale theory.
We derive the exponential as well as power decreasing tail estimations for normed sums of centered independent identical distributed (or not) random variables on the Khintchine's form. We consider arbitrary, in particular, non-Rademacher's variables and not only Lebesgue-Riesz rearrangement invariant norms for the random variables. We intend to calculate the value of correspondent limit.
We study sufficient conditions for the belonging of random process to certain Besov space and for the Central Limit Theorem (CLT) in these spaces. We investigate also the non-asymptotic tail behavior of normed sums of centered random independent variables (vectors) with values in these spaces. Main apparatus is the theory of mixed (anisotropic) Lebesgue-Riesz spaces, in particular so-called permutation inequality.
We calculate the so-called Rademacher's Grand Lebesgue Space norm for a centered (shifted) indicator (Bernoulli's, binary) random variable. This norm is optimal for the centered and bounded random variables (r.v.). Using this result we derive a very simple bilateral sharp exponential tail estimates for sums of these variables, not necessary to be identical distributed, under non-standard norming, and give some examples to show the exactness of our estimates.
Let Ξbe a discrete set in R^d. Call the elements of Ξcenters. The well-known Voronoi tessellation partitions R^d into polyhedral regions (of varying volumes) by allocating each site of R^d to the closest center. Here we study allocations of R^d to Ξin which each center attempts to claim a region of equal volume α. We focus on the case where Ξarises from a Poisson process of unit intensity. It was proved in math.PR/0505668 that there is a unique allocation which is stable in the sense of the Gale-Shapley marriage problem. We study the distance X from a typical site to its allocated center in th…
We construct an example of a continuous centered random process with light tails of finite-dimensional distribution but with (relatively) heavy tail of maximum distribution. The apparatus for tails comparison are embedding results for Orlicz and Grand Lebesgue Spaces (GLS).
We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of random variables.
We generalize a famous tail Doob's inequality, relative two non-negative random variables, arising in the martingale theory, in two directions: on the more general source data and on the random variables belonging to the so-called Grand Lebesgue Spaces. We bring also several examples in each sections in order to show the exactness of our estimates.
We present in this paper the Banach space representation for the set of random finite-dimensional vectors with exponential decreasing tails of distributions. We show that there are at last three types of these multidimensional Banach spaces, i.e. which can completely describe the random vectors with exponential decreasing tails of distributions: exponential Orlicz spaces, Young spaces and Grand Lebesgue spaces. We discuss in the last section the possible applications of obtained results.
We deduce the non-asymptotical (bilateral) estimates for moment inequalities for multiple sums of non-negative (more precisely, non-negative) independent random variables, on the other words, the well known U or V-statistics. Our consideration based on the correspondent estimates for the one-dimensional case by means of the so-called degenerate approximation. We apply also the theory of Bell functions as well as the properties of the Poisson distribution and the theory of the so-called Grand Lebesgue Spaces (GLS).
For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail…
We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as t…
This paper develops asymptotic approximations of $P(\int_Te^{f(t)}\,dt>b)$ as $b\rightarrow\infty$ for a homogeneous smooth Gaussian random field, $f$, living on a compact $d$-dimensional Jordan measurable set $T$. The integral of an exponent of a Gaussian random field is an important random variable for many generic models in spatial point processes, portfolio risk analysis, asset pricing and so forth. The analysis technique consists of two steps: 1. evaluate the tail probability $P(\int_Ξe^{f(t)}\,dt>b)$ over a small domain $Ξ$ depending on $b$, where $\operatorname {mes}(Ξ)\rightarrow0$ as …
Exact exponential tail estimation for sums of independent centered random variables, under natural norming, with applications to the theory of U-statistics
本文利用 GLS 理论推导自然规范化独立中心化随机变量之和的精确指数尾部，并给出 U 统计量中的精细尾部估计。
