The monotone rearrangement of a function is the non-decreasing function with the same distributio... more The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics. A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the Fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.
Main results of this paper were obtained together with Alexander Nagaev, with whom the first auth... more Main results of this paper were obtained together with Alexander Nagaev, with whom the first author had collaborated for more than 35 years, until Alexander's tragic death in 2005. Since then, we have gathered strength and finalised this paper, strongly feeling Alexander's absence -our memories of him will stay with us forever.
The article discusses the asymptotic behavior of a particle performing so-called "random flight."... more The article discusses the asymptotic behavior of a particle performing so-called "random flight." In a recent work by , when the moments T k of changing the direction of the particle form an inhomogeneous Poisson process, it was shown that, depending on the nature of the inhomogeneity, three variants of the limiting distribution arise naturally for the zoomed particle trajectory. The purpose of this work is to show that these three options are preserved under much more general assumptions about the sequence (T k ). Bibliography: 10 titles.
Convexity plays a prominent role in a number of problems, but practical considerations frequently... more 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.
Springer Proceedings in Mathematics & Statistics, 2017
In the first part of the paper, we consider a "random flight" process in R d and obtain the weak ... more In the first part of the paper, we consider a "random flight" process in R d and obtain the weak limits under different transformations of the Poissonian switching times. In the second part, we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result, we use the approach of . We consider more general model which may be called "random walk over ellipsoids in R d ". For this model, we establish the Edgeworthtype expansion. The main tool in this part is the parametrix method .
We introduce a modification of the index of increase that works in both deterministic and random ... more We introduce a modification of the index of increase that works in both deterministic and random environments, and thus allows us to assess monotonicity of functions that are prone to random measurement errors. We prove consistency of the empirical index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results allow us to determine the frequency at which observations should be taken in order to reach any pre-specified level of estimation precision. We illustrate the performance of the suggested estimator using simulated data arising from purely deterministic and error-contaminated monotonic and non-monotonic functions.
We consider the moving particle process in R which is defined in the following way. There are two... more We consider the moving particle process in R which is defined in the following way. There are two independent sequences (Tk) and (εk) of random variables. The variables Tk are non negative and ∀k Tk ≤ Tk+1, while variables εk form an i.i.d sequence with common distribution concentrated on the unit sphere Sd−1. The values εk are interpreted as the directions, and Tk as the moments of change of directions. A particle starts from zero and moves in the direction ε1 up to the moment T1. It then changes direction to ε2 and moves on within the time interval T2 − T1, etc. The speed is constant at all sites. The position of the particle at time t is denoted by X(t). Study of the processes of this type has a long history. The first work dates back probably to Pearson and continued by Kluyer (1906) and Rayleigh (1919). Mandelbrot (1982) considered the case where the increments Tn − Tn−1 form i.i.d. sequence with the common law having a heavy tail. He also introduced the term ”Levy flights” lat...
In this paper we introduce 5 natural definitions (AI-0) - (AI-4) of asymptotic independence (AI) ... more In this paper we introduce 5 natural definitions (AI-0) - (AI-4) of asymptotic independence (AI) of two sequences ($X_n$), ($Y_n$) 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 AI-0 - AI-4, we provide some counterexamples.
Proceedings of the American Mathematical Society, 2003
We consider weak convergence of empirical measures generated by stationary random process X X per... more We consider weak convergence of empirical measures generated by stationary random process X X perturbed by deterministic noise N N . We assume that the noise N N has asymptotic distribution. In particular, we demonstrate that if the process X X is ergodic, or satisfies some mixing assumptions, then the influence of deterministic noise N N on X X is the same as it would be if N N were stochastic. Such results are of importance when investigating fluctuations and convex rearrangements of stochastic processes.
In the previous work by the second author (A. Tempelman, Randomized multivariate central limit th... more In the previous work by the second author (A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications, 143 (2022), 89-105) CLTs for homogeneous random fields on R m and Z m (m ≥ 1), satisfying the condition: E[|X(0)| 2+δ ] < ∞ have been proved. In the present article the authors prove that these theorems are valid under the weaker condition E[(X(0)) 2 ] < ∞.
Summary In recent decades, substantial changes have been observed in the left and right tails of ... more Summary In recent decades, substantial changes have been observed in the left and right tails of income distributions in countries like the USA, Germany, the UK, and France. These changes are a major concern for policy makers. Here, we derive inferential results for a new inequality index that is specifically designed for capturing such significant shifts. We propose two empirical estimators for the index and show that they are asymptotically equivalent. Afterward, we adopt one estimator and prove its consistency and asymptotic normality. Finally, we introduce an empirical estimator for its variance and provide conditions for its consistency. An analysis of real data from the Bank of Italy Survey of Income and Wealth is also presented on the basis of the obtained inferential results.
We introduce and explore an empirical index of increase that works in both deterministic and rand... more 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.
In various research areas related to decision making, problems and their solutions frequently rel... more 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.
It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d , its ... more It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d , its convex hull V (t) = conv{ B(s), s ≤ t} with probability 1 for each t > 0 contains 0 as an interior point (see ). We also know that the winding number of a typical path of a 2-dimensional BM is equal to +∞. 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.
Soit X un vecteur aléatoire dans ℝ d ayant la queue à variation régulière avec l’exposant α et la... more Soit X un vecteur aléatoire dans ℝ d ayant la queue à variation régulière avec l’exposant α et la mesure spectrale σ, et soit (|X|,(X/(|X|))) son décomposition polaire. L’auteurs montrent que si f:S d-1 →S d-1 est σ-p.p. continue, alors le vecteur aléatoire (|X|,f((X/(|X|)))) a la queue à variation régulière d’exposant α et de mesure spectrale σf -1 . Si h:S d-1 →R * est σ-p.p. continue et bornée, ils montrent que le vecteur aléatoire (|X|h((X/(|X|))),(X/(|X|))) a la queue à variation régulière d’exposant α et de mesure spectrale h α σ.Reviewer: Aurel Spataru (Bucuresti)
The objective of this paper is to extend an estimation method of parameters of the stable distrib... more The objective of this paper is to extend an estimation method of parameters of the stable distributions in $\rd$ to the regularly varying tails distributions in an arbitrary cone. The consistency and the asymptotic normality of estimators are proved. The sampling method of regrouping is modified to optimize the rate of convergence of estimators.
We consider the problem of the convergence of the so-called Le Page series in the Skohorod space ... more 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 $\alpha$-stable distribution on $\bbD^d$ with given spectral measure.
Хорошо известно, что для последовательности независимых одинаково распределенных случайных величи... more Хорошо известно, что для последовательности независимых одинаково распределенных случайных величин соответствующие нормированные ступенчатые процессы слабо сходятся к винеровскому процессу. Более сильная сходимость, а именно, сходимость по вариации распределений функционалов от этих процессов, была установлена в [4] в предположении конечности информации Фишера для случайных величин. В настоящей статье мы доказываем такую сходимость без каких-либо условий, связанных с информацией Фишера. Ключевые слова и фразы: принципы инвариантности, сходимость по вариации, локальные предельные теоремы.
ABSTRACT We consider random linear processes and fields with continuous time or space argument an... more ABSTRACT We consider random linear processes and fields with continuous time or space argument and demonstrate how the analogue of Beveridge-Nelson decomposition can be applied to prove limit theorems for integrals of such processes and fields.
The monotone rearrangement of a function is the non-decreasing function with the same distributio... more The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics. A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the Fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.
The monotone rearrangement of a function is the non-decreasing function with the same distributio... more The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics. A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the Fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.
Main results of this paper were obtained together with Alexander Nagaev, with whom the first auth... more Main results of this paper were obtained together with Alexander Nagaev, with whom the first author had collaborated for more than 35 years, until Alexander's tragic death in 2005. Since then, we have gathered strength and finalised this paper, strongly feeling Alexander's absence -our memories of him will stay with us forever.
The article discusses the asymptotic behavior of a particle performing so-called "random flight."... more The article discusses the asymptotic behavior of a particle performing so-called "random flight." In a recent work by , when the moments T k of changing the direction of the particle form an inhomogeneous Poisson process, it was shown that, depending on the nature of the inhomogeneity, three variants of the limiting distribution arise naturally for the zoomed particle trajectory. The purpose of this work is to show that these three options are preserved under much more general assumptions about the sequence (T k ). Bibliography: 10 titles.
Convexity plays a prominent role in a number of problems, but practical considerations frequently... more 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.
Springer Proceedings in Mathematics & Statistics, 2017
In the first part of the paper, we consider a "random flight" process in R d and obtain the weak ... more In the first part of the paper, we consider a "random flight" process in R d and obtain the weak limits under different transformations of the Poissonian switching times. In the second part, we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result, we use the approach of . We consider more general model which may be called "random walk over ellipsoids in R d ". For this model, we establish the Edgeworthtype expansion. The main tool in this part is the parametrix method .
We introduce a modification of the index of increase that works in both deterministic and random ... more We introduce a modification of the index of increase that works in both deterministic and random environments, and thus allows us to assess monotonicity of functions that are prone to random measurement errors. We prove consistency of the empirical index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results allow us to determine the frequency at which observations should be taken in order to reach any pre-specified level of estimation precision. We illustrate the performance of the suggested estimator using simulated data arising from purely deterministic and error-contaminated monotonic and non-monotonic functions.
We consider the moving particle process in R which is defined in the following way. There are two... more We consider the moving particle process in R which is defined in the following way. There are two independent sequences (Tk) and (εk) of random variables. The variables Tk are non negative and ∀k Tk ≤ Tk+1, while variables εk form an i.i.d sequence with common distribution concentrated on the unit sphere Sd−1. The values εk are interpreted as the directions, and Tk as the moments of change of directions. A particle starts from zero and moves in the direction ε1 up to the moment T1. It then changes direction to ε2 and moves on within the time interval T2 − T1, etc. The speed is constant at all sites. The position of the particle at time t is denoted by X(t). Study of the processes of this type has a long history. The first work dates back probably to Pearson and continued by Kluyer (1906) and Rayleigh (1919). Mandelbrot (1982) considered the case where the increments Tn − Tn−1 form i.i.d. sequence with the common law having a heavy tail. He also introduced the term ”Levy flights” lat...
In this paper we introduce 5 natural definitions (AI-0) - (AI-4) of asymptotic independence (AI) ... more In this paper we introduce 5 natural definitions (AI-0) - (AI-4) of asymptotic independence (AI) of two sequences ($X_n$), ($Y_n$) 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 AI-0 - AI-4, we provide some counterexamples.
Proceedings of the American Mathematical Society, 2003
We consider weak convergence of empirical measures generated by stationary random process X X per... more We consider weak convergence of empirical measures generated by stationary random process X X perturbed by deterministic noise N N . We assume that the noise N N has asymptotic distribution. In particular, we demonstrate that if the process X X is ergodic, or satisfies some mixing assumptions, then the influence of deterministic noise N N on X X is the same as it would be if N N were stochastic. Such results are of importance when investigating fluctuations and convex rearrangements of stochastic processes.
In the previous work by the second author (A. Tempelman, Randomized multivariate central limit th... more In the previous work by the second author (A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications, 143 (2022), 89-105) CLTs for homogeneous random fields on R m and Z m (m ≥ 1), satisfying the condition: E[|X(0)| 2+δ ] < ∞ have been proved. In the present article the authors prove that these theorems are valid under the weaker condition E[(X(0)) 2 ] < ∞.
Summary In recent decades, substantial changes have been observed in the left and right tails of ... more Summary In recent decades, substantial changes have been observed in the left and right tails of income distributions in countries like the USA, Germany, the UK, and France. These changes are a major concern for policy makers. Here, we derive inferential results for a new inequality index that is specifically designed for capturing such significant shifts. We propose two empirical estimators for the index and show that they are asymptotically equivalent. Afterward, we adopt one estimator and prove its consistency and asymptotic normality. Finally, we introduce an empirical estimator for its variance and provide conditions for its consistency. An analysis of real data from the Bank of Italy Survey of Income and Wealth is also presented on the basis of the obtained inferential results.
We introduce and explore an empirical index of increase that works in both deterministic and rand... more 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.
In various research areas related to decision making, problems and their solutions frequently rel... more 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.
It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d , its ... more It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d , its convex hull V (t) = conv{ B(s), s ≤ t} with probability 1 for each t > 0 contains 0 as an interior point (see ). We also know that the winding number of a typical path of a 2-dimensional BM is equal to +∞. 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.
Soit X un vecteur aléatoire dans ℝ d ayant la queue à variation régulière avec l’exposant α et la... more Soit X un vecteur aléatoire dans ℝ d ayant la queue à variation régulière avec l’exposant α et la mesure spectrale σ, et soit (|X|,(X/(|X|))) son décomposition polaire. L’auteurs montrent que si f:S d-1 →S d-1 est σ-p.p. continue, alors le vecteur aléatoire (|X|,f((X/(|X|)))) a la queue à variation régulière d’exposant α et de mesure spectrale σf -1 . Si h:S d-1 →R * est σ-p.p. continue et bornée, ils montrent que le vecteur aléatoire (|X|h((X/(|X|))),(X/(|X|))) a la queue à variation régulière d’exposant α et de mesure spectrale h α σ.Reviewer: Aurel Spataru (Bucuresti)
The objective of this paper is to extend an estimation method of parameters of the stable distrib... more The objective of this paper is to extend an estimation method of parameters of the stable distributions in $\rd$ to the regularly varying tails distributions in an arbitrary cone. The consistency and the asymptotic normality of estimators are proved. The sampling method of regrouping is modified to optimize the rate of convergence of estimators.
We consider the problem of the convergence of the so-called Le Page series in the Skohorod space ... more 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 $\alpha$-stable distribution on $\bbD^d$ with given spectral measure.
Хорошо известно, что для последовательности независимых одинаково распределенных случайных величи... more Хорошо известно, что для последовательности независимых одинаково распределенных случайных величин соответствующие нормированные ступенчатые процессы слабо сходятся к винеровскому процессу. Более сильная сходимость, а именно, сходимость по вариации распределений функционалов от этих процессов, была установлена в [4] в предположении конечности информации Фишера для случайных величин. В настоящей статье мы доказываем такую сходимость без каких-либо условий, связанных с информацией Фишера. Ключевые слова и фразы: принципы инвариантности, сходимость по вариации, локальные предельные теоремы.
ABSTRACT We consider random linear processes and fields with continuous time or space argument an... more ABSTRACT We consider random linear processes and fields with continuous time or space argument and demonstrate how the analogue of Beveridge-Nelson decomposition can be applied to prove limit theorems for integrals of such processes and fields.
The monotone rearrangement of a function is the non-decreasing function with the same distributio... more The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics. A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the Fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.
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