arXiv subjects

Youri Davydov

Publications and source records attributed to Youri Davydov.

At least 19 recordsLinked to original sources

arXiv2026

A counter-example linked to Gaussian convex hulls

We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov- Paulauskas (2024), its normalized convex hulls converge, with probability 1, to the concentration ellipsoid of the limiting distribution. The goal of the present note is to show that if the assumption of weak convergence of the initial sequence is relaxed, than the limit set can be an arbitrary convex compact set.

math.PR↗

arXiv2025

Point Process Approach to the Winner Problem

We consider a limit theorem for a triangular array of point processes generated by non-identically distributed random variables, and apply the result for the analysis of the limiting behavior of the Argmaximum of independent random variables, as well as for some step processes.

math.PR↗

arXiv2024

The Distribution of Argmaximum or a Winner Problem

We consider a limit theorem for the distribution of a r.v. $Y_n:=argmax {\{X_i, i= 1,..., n\}},$ where $X_i'$s are independent continuous non-negative random variables. The r.v.'s $\{X_i, i=1,..., n\}$, may be interpreted as the gains of $n$ players in a game, and the r.v. $Y_n$ itself as the number of a ``winner". In the case of i.i.d.r.v.'s, the distribution of $Y_n$ is, clearly, uniform on $\{1,..., n\},$ while when the $X'$s are non-identically distributed, the problem requires some calculations.

math.PR↗

arXiv2022

Randomized limit theorems for stationary ergodic random processes and fields

We consider "randomized" statistics constructed by using a finite number of observations a random field at randomly chosen points. We generalize the invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the theorem about convergence to the Brownian bridge and the Kolmogorov theorem about the limit distribution of the empirical distribution function, as well as an improved version of the CLT in A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89-105. The randomized approach, introduced in the mentioned work, allows to extend these theorems to all ergodic homogeneous random fields on $\Z^m$ and $\R^m.$

math.PR↗

arXiv2020

Remarks on asymptotic independence

In this paper we introduce several natural definitions of asymptotic independence of two sequences of random elements. We discuss their basic properties, some simple connections between them and connections with properties of weak dependence. In particular, the case of tight sequences is considered in detail. Finally, in order to clarify the relationships between different definitions, we provide some counterexamples.

math.PR↗

arXiv2020

More on the convergence of Gaussian convex hulls

A "law of large numbers" for consecutive convex hulls for weakly dependent Gaussian sequences $\{X_n\}$, having the same marginal distribution, is extended to the case when the sequence $\{X_n\}$ has a weak limit. Let $\mathbb{B}$ be a separable Banach space with a conjugate space $\mathbb{B}^\ast$. Let $\{X_n\}$ be a centered $\mathbb{B}$-valued Gaussian sequence satisfying two conditions: 1) $X_n \Rightarrow X\;\;$ and 2) For every $x^* \in \mathbb{B}^\ast$ $$ \lim_ {n,m, |n-m|\rightarrow \infty}E\langle X_n, x^*\rangle \langle X_m, x^*\rangle\;\; = \;\;0. $$ Then with probability 1 the normalized convex hulls $$ W_n = \frac{1}{(2\ln n)^{1/2}}\,{\rm conv} \{\,X_1,\ldots,X_{n}\,\} $$ converge in Hausdorff distance to the concentration ellipsoid of a limit Gaussian $\mathbb{B}$-valued random element $X.$ In addition, some related questions are discussed.

math.PR↗

arXiv2018

Searching for, and quantifying, non-convexity of functions

Convexity plays a prominent role in a number of problems, but practical considerations frequently give rise to non-convex functions. We suggest a method for determining convex regions, and also for assessing the lack of convexity in the other regions. The method relies on a specially constructed decomposition of symmetric matrices, such as the Hessian. We illustrate theoretical results using several examples, one of which analyses a problem arising in risk measurement and management in insurance and finance.

math.FA↗

arXiv2018

Estimating the index of increase via balancing deterministic and random data

We introduce and explore an empirical index of increase that works in both deterministic and random environments, thus allowing to assess monotonicity of functions that are prone to random measurement-errors. We prove consistency of the index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results suggest a frequency at which observations should be taken in order to reach any pre-specified level of estimation precision. We illustrate the index using data arising from purely deterministic and error-contaminated functions, which may or may not be monotonic.

math.ST↗

arXiv2017

Quantifying non-monotonicity of functions and the lack of positivity in signed measures

In various research areas related to decision making, problems and their solutions frequently rely on certain functions being monotonic. In the case of non-monotonic functions, one would then wish to quantify their lack of monotonicity. In this paper we develop a method designed specifically for this task, including quantification of the lack of positivity, negativity, or sign-constancy in signed measures. We note relevant applications in Insurance, Finance, and Economics, and discuss some of them in detail.

math.PR↗

arXiv2017

On distance in total variation between image measures

We are interested in the estimation of the distance in total variation $$ Δ:= \|P_{f(X)} - P_{g(X)}\|_{\mathrm var} $$ between distributions of random variables $f(X)$ and $g(X)$ in terms of proximity of $f$ and $g.$ We propose a simple general method of estimating $Δ$. For Gaussian and trigonometrical polynomials it gives an asymptotically optimal result (when the degree tends to $\infty$).

math.PR↗

arXiv2017

Inferential results for a new measure of inequality

In this paper we derive inferential results for a new index of inequality, specifically defined for capturing significant changes observed both in the left and in the right tail of the income distributions. The latter shifts are an apparent fact for many countries like US, Germany, UK, and France in the last decades, and are a concern for many policy makers. We propose two empirical estimators for the index, and show that they are asymptotically equivalent. Afterwards, we adopt one estimator and prove its consistency and asymptotic normality. Finally we introduce an empirical estimator for its variance and provide conditions to show its convergence to the finite theoretical value. An analysis of real data on net income from the Bank of Italy Survey of Income and Wealth is also presented, on the base of the obtained inferential results.

math.ST↗

arXiv2017

Lamperti type theorems for random fields

In the paper we consider Lamperti type theorems for random fields. Together with known results we present some new results on ${\mathbb R}^m$-valued self-similar fields $\{{\bf X} ({\bf t}), \ {\bf t} \in {\mathbb R}^d \}$, their domains of attraction and the so-called Lamperti transformations, expressing the relation between self-similarity and stationarity. Also we investigate regularly and slowly varying functions of several variables.

math.PR↗

arXiv2016

Random walks in non homogeneous Poissonian environment

We consider the moving particle process in Rd which is defined in the following way. There are two independent sequences (Tk) and (dk) of random variables. The variables Tk are non negative and form an increasing sequence, while variables dk form an i.i.d sequence with common distribution concentrated on the unit sphere. The values dk are interpreted as the directions, and Tk as the moments of change of directions. A particle starts from zero and moves in the direction d1 up to the moment T1 . It then changes direction to d2 and moves on within the time interval T2 minus T1 , etc. The speed is constant at all sites. The position of the particle at time t is denoted by X(t). We suppose that the points (Tk) form a non homogeneous Poisson point process and we are interested in the global behavior of the process (X(t)), namely, we are looking for conditions under which the processes (Y(T,t), T is non negative), Y(T,t) is X(tT) normalized by B(T), t in (0, 1), weakly converges in C(0, 1) to some process Y when T tends to infinity. In the second part of the paper the process X(t) is considered as a Markov chain. We construct diffusion approximations for this process and investigate their accuracy. The main tool in this part is the paramertix method.

math.PR↗

arXiv2015

On convex hull and winding number of self similar processes

It is well known that for a standard Brownian motion (BM) $ \{B(t), \;t \geq 0\}$ with values in $\mathbb{R}^d$, its convex hull $ V(t)=\conv \{\{\,B(s),\;s \leq t \}$ with probability $1$ for each $t > 0$ contains $0$ as an interior point (see Evans (1985)). We also know that the winding number of a typical path of a $2$-dimensional BM is equal to $+\infty.$ The aim of this article is to show that these properties aren't specifically "Brownian", but hold for a much larger class of $d$-dimensional self similar processes. This class contains in particular $d$-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Levy processes.

math.PR↗

arXiv2012

On the asymptotic of convex hulls of Gaussian fields

We consider a Gaussian field $X = \{X_t, t \in T\}$ with values in a Banach space $B$ defined on a parametric set $T$ equal to $R^m$ or $Z^m.$ It is supposed that the distribution $\cal P$ of $X_t$ is independent of $t.$ We consider the asymptotic behavior of closed convex hulls $$ W_n = \conv \{X_t, t \in T_n\} $$ where $(T_n)$ is an increasing sequence of subsets of $T$ and we show that under some conditions of the weak dependence with probability 1 $$ \lim_{n\rightarrow \infty} \frac{1}{b_n}\,W_n = {\cal E} $$ (in the sense of Hausdorff distance), where the limit shape ${\cal E}$ is the concentration ellipsoid of $\cal P.$ The asymptotic behavior of the mathematical expectations $Ef(W_n),$ where $f$ is an homogeneous function is also studied.

math.PR↗

arXiv2011

Stability for random measures, point processes and discrete semigroups

Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly defining the scaling operation as thinning of counting measures we characterise the corresponding discrete stability property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. We give spectral and LePage representations for general strictly stable random measures without assuming their independent scattering. As a consequence, spectral representations are obtained for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for such processes is also derived using the so-called Sibuya point processes, which constitute a new family of purely random point processes. The obtained results are then applied to explore stable random elements in discrete semigroups, where the scaling is defined by means of thinning of a point process on the basis of the semigroup. Particular examples include discrete stable vectors that generalise discrete stable random variables and the family of natural numbers with the multiplication operation, where the primes form the basis.

math.PR↗

arXiv2011

On the convergence of Le Page series in Skohorod space

We consider the problem of the convergence of the so-called Le Page series in the Skohorod space $\bbD^d=\bbD([0,1],\bbR^d)$ and provide a simple criterion based on the moments of the increments of the random process involved in the series. This provides a simple sufficient condition for the existence of an $α$-stable distribution on $\bbD^d$ with given spectral measure.

math.PR↗

arXiv2011

On convex hull of d-dimensional fractional Brownian motion

It is well known that for standard Brownian motion $ \{B(t), \;t \geq 0\}$ with values in $\mathbb{R}^d$ its convex hull $ V(t)=\conv \{\{\,B(s),\;s \leq t \}$ with probability 1 contains 0 as an interior point for each $t > 0$ (see \cite{E}). The aim of this note is to state the analoguos property for $d$-dimensional fractional Brownian motion.

math.PR↗