Distribution of functionals of bridges for diffusions with jumps

The paper deals with a method of calculation of distributions for functionals of bridges of a process which is a generalization of a diffusion with jumps. The approach to calculation of distributions for integral functionals of bridges is the same as for the diffusion itself. This approach is based on calculation of the Laplace transform of distributions of the integral functionals. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 34–47.     

Asymptotic normality of spherical means of nonlinear functionals of Gaussian random fields

The central limit theorem is proved for the integral-type functionals of nonlinear transformations of two-and three-dimensional uniform isotropic Gaussian random fields. A theorem on convergence of finite-dimensional distributions of these functionals to the corresponding distributions of the Wiener process is also established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 472–480, April, 1993.     

Random locations of periodic stationary processes

We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that the set of all possible distributions of intrinsic location functionals for periodic stationary processes is the convex hull generated by a specific group of distributions. We then focus on two special subclasses of these random locations. For the first subclass, the density has a uniform lower bound; for the second subclass, the possible distributions are closely related to the concept of joint mixability.     

广义维纳泛函,白噪声分布及实Schwartz广义函数空间上的分布

在实Ｓｃｈｗａｒｔｚ广义函数空间上，证明了复值广义维纳泛函，由Ｋｏｎｄｒａｔｅｖ－Ｓｔｒｅｉｔ及Ｈｉｄａ构造的复值白噪声分布都是由Ｋｈｒｅｎｎｉｋｏｖ构造的分布。利用上述结果进而证明了，一类无穷维伪微分算子是由复值广义维纳泛函空间上的连续线性算子族扩张而成。更进一步，还证明了由Ｋｈｒｅｎｎｉｋｏｖ构造的关于分布的试验函数空间是关于白噪声泛函的Ｍｅｙｅｒ－Ｙａｎ试验函数空间的子空间。     

Distribution of functionals of Brownian motion stopped at a time that is inverse to local time

Integral functionals of Brownian motion and of Brownian local time, as well as the supremum of Brownian motion and the supremum of Brownian local time are considered. The obtained results allow the computation of the distributions of these functionals for a Brownian motion stopped at the moment when the local time attains first a given value at one of two levels. It has been established that for this stopping time the Brownian local time is a Markov process with respect to the space variable and the generating operator of the process has been found. Examples of the computation of the distributions of certain functionals are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 37–61, 1990.     

Distributions of Some Functionals of the Brownian Local Time

Some functionals of the Brownian local time are considered. For these functionals, we propose methods of computation of distributions of the functionals. Bibliography: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 22–35.     

On Distribution of Functionals of a Brownian Motion Stopped at the Inverse Range Time

In this paper, we develop methods for computation of distributions of functionals of a Brownian motion stopped at the inverse range time. Some explicit formulas for distributions of functionals are obtained. Bibliography: 7 titles.     

Complete asymptotic expansions for lower (upper) continuous random walks with two boundaries. I

The work is devoted to the investigation of some asymptotic properties of probability distributions generated by semicontinuous random walks. Complete asymptotic expansions of the distributions of the boundary functionals connected with two boundaries (a walk in a strip) are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 64–101, 1976.     

Lorentz-Covariant Ultradistributions, Hyperfunctions, and Analytic Functionals

We generalize the theory of Lorentz-covariant distributions to broader classes of functionals including ultradistributions, hyperfunctions, and analytic functionals with a tempered growth. We prove that Lorentz-covariant functionals with essential singularities can be decomposed into polynomial covariants and establish the possibility of the invariant decomposition of their carrier cones. We describe the properties of odd highly singular generalized functions. These results are used to investigate the vacuum expectation values of nonlocal quantum fields with an arbitrary high-energy behavior and to extend the spin–statistics theorem to nonlocal field theory.     

A Computational Approach for Full Nonparametric Bayesian Inference Under Dirichlet Process Mixture Models

Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.     

Absolute continuity of the images of measures

One presents a new method for isolating conditions for the absolute continuity of distributions of functionals of random processes. The obtained general result is applied to the investigation of integral functionals of stationary processes, to the supremum of semistable processes, and to the norm of stable Banach-valued vectors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 48–54, 1985.     

On bounded maximum width sequential confidence ellipsoids based on generalized U-statistics

For a vector of (estimable) functionals of several independent distributions, sequential confidence ellipsoids (of bounded maximum width) based on a class of generalized U-statistics are studied. A stopping rule along with a procedure for choosing the component sample sizes at each stage is developed, so that the proposed confidence ellipsoid has a confidence coefficient asymptotically (as the prescribed maximum width shrinks to zero) equal to a preassigned 1 - α (0 < α < 1), and the expected total sample size is minimized for the procedure. Asymptotic efficiency of the procedure is also studied. The case of von Mises' functionals is treated briefly at the end.     

Boundary Functionals of a Semicontinuous Process with Independent Increments on an Interval

We investigate boundary functionals of a semicontinuous process with independent increments on an interval with two reflecting boundaries. We determine the transition and ergodic distributions of the process, as well as the distributions of boundary functionals of the process, namely, the time of first hitting the upper (lower) boundary, the number of hittings of the boundaries, the number of intersections of the interval, and the total sojourn time of the process on the boundaries and inside the interval. We also present a limit theorem for the ergodic distribution of the process and asymptotic formulas for the mean values of the distributions considered.     

Nondifferentiable optimization via smooth approximation: General analytical approach

In this paper we present a method for nondifferentiable optimization, based on smoothed functionals which preserve such useful properties of the original function as convexity and continuous differentiability. We show that smoothed functionals are convenient for implementation on computers. We also show how some earlier results in nondifferentiable optimization based on smoothing-out of kink points can be fitted into the framework of smoothed functionals. We obtain polynomial approximations of any order from smoothed functionals with kernels given by Beta distributions. Applications of smoothed functionals to optimization of min-max and other problems are also discussed.     

Distribution of integral functionals of the local time of a Bessel process

In the paper one considers methods allowing one to determine the distribution of certain functionals of the local times of diffusion processes. These methods turn out to be most effective for the Brownian local time and for the local time of a third-order Bessel process. For Brownian local time these methods have been obtained earlier and the main part of the paper is devoted to the development of methods for the evaluation of the distributions of integral functionals and functionals of the supremum type of the local time of a third-order Bessel process.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 8–27, 1989.     

Joint distributions of random linear functionals

An estimate is given of the mean value with respect to a Gaussian measure of the distance between distributions of two finite collections of linear functionals over a Euclidean space with probability measure for certain metrics in the space of finite-dimensional distributions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 115–121, 1987.     

An excursion approach to maxima of the Brownian bridge

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov–Smirnov statistic and the Kuiper statistic.     

Distributions of functionals of the Brownian motion stopped at the moment inverse to the sojourn time

The paper deals with methods of computing the distributions of functionals of the Brownian motion stopped at some random moments. The moments are obtained by means of the minimum and maximum operations from the moments inverse to some additive functionals. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 39–56.     

Distributions of Additive Functionals of Brownian Motion Stopped at Moments of Maxima and Minima of Random Times

We study distributions of additive functionals of a Brownian motion stopped at moments constructed by maxima and minima operations from the inverse range time, moments inverse to some additive functionals, and the first exit time. Some interesting examples of application are considered. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 36–53.     

Distributions of functionals of diffusions with jumps

The paper deals with methods of computing the distributions of functionals of a process that is a diffusion with jumps occurring according to a compound Poisson process. For symmetric processes, some exact formulas for distributions related to the first exit time are derived. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 27–41.