Leonid Tolmatz - Academia.edu (original) (raw)

Papers by Leonid Tolmatz

["The exact tail asymptotics of the distribution of the functional {dollar}{bsol}rm {bsol}in... more ["The exact tail asymptotics of the distribution of the functional {dollar}{bsol}rm {bsol}int{bsol}sbsp{lcub}o{rcub}{lcub}1{rcub}{bsol}vert B{bsol}sb{lcub}s{rcub}{bsol}vert ds{dollar} on the Wiener process {dollar}{bsol}rm {bsol}{lcub}B{bsol}sb{lcub}s{rcub}{dollar}: {dollar}{bsol}rm S{bsol}ge o{bsol}{rcub}{dollar} is obtained."]

Discrete Mathematics & Theoretical Computer Science, 2003

The distribution function of the integral of the absolute value of the Brownian motion was expres... more The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

Statistics & Probability Letters, Aug 1, 2004

Let F be a distribution function with all moments finite and such that the problem of moments for... more Let F be a distribution function with all moments finite and such that the problem of moments for F has a nonunique solution (F is M-indeterminate). Our goal is to explicitly describe a Stieltjes class S ¼ ff e ¼ f ½1 þ eh; e 2 ½À1; 1g of distributions (here written in terms of the densities) all having the same moments as F. We study in detail the case when F is the distribution of the power transformation x r ; r40 of a random variable x with generalized gamma distribution. We derive new Stieltjes classes in this case and also for powers of the normal and the exponential distributions. We find the value of the index of dissimilarity for some of these classes.

Journal of Mathematical Analysis and Applications, Apr 1, 2005

The double Laplace transform of the distribution function of the integral of the positive part of... more The double Laplace transform of the distribution function of the integral of the positive part of the Brownian bridge was determined by M. Perman and J.A. Wellner, as well as the moments of this distribution. The purpose of the present paper is to determine the asymptotics of this distribution for large values of the argument, and the corresponding asymptotics of the moments.

Annals of Probability, 2002

Smirnov obtained the distribution F for his ω 2-test in the form of a certain series. F is identi... more Smirnov obtained the distribution F for his ω 2-test in the form of a certain series. F is identical to the distribution of the the Brownian bridge in the L 2 norm. Smirnov, Kac and Shepp determined the Laplace-Stieltjes transform of F. Anderson and Darling expressed F in terms of Bessel functions. In the present paper we compute the moments of F and their asymptotics, obtain expansions of F and its density f in terms of the parabolic cylinder functions and Laguerre functions, and determine their asymptotics for the small and large values of the argument. A novel derivation of expansions of Smirnov and of Anderson and Darling is obtained.

arXiv (Cornell University), Apr 12, 2009

The density distribution function of the integral of the exponential Brownian motion is determine... more The density distribution function of the integral of the exponential Brownian motion is determined explicitly in the form of a rapidly convergent series. 1 F 1 (α; γ; z) is the confluent hypergeometric function,

Annals of Probability, 2003

The Annals of Probability (2002) 30 253-269 EDITOR'S NOTE. The following Note Added in Proof and ... more The Annals of Probability (2002) 30 253-269 EDITOR'S NOTE. The following Note Added in Proof and tables were inadvertently omitted from the original publication. The editorial staff regrets the error. NOTE ADDED IN PROOF. The author has recently learned, following a personal communication with J. Pitman, that the Laplace inversion in Theorem 2 is already known in the literature; see J. Kiefer, K-sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests, Annals of Mathematical Statistics (1959) 30 420-447. The latter paper also contains tables of the distribution function F (λ); however, Table 4 in the present paper provides more accurate values in certain entries.

Applied Mathematics and Computation, Jun 1, 2005

Let F be a distribution function with density f, finite moments of any positive integer order and... more Let F be a distribution function with density f, finite moments of any positive integer order and such that the problem of moments for F has a nonunique solution (F is Mindeterminate). We are looking for explicit Stieltjes classes S = {f e = f [1 + eh], e 2 [À1, 1]} of distributions (here in terms of their densities) all sharing the same moments as those of F. We suggest a general method for constructing such classes. Then we apply our method to describe the M-indeterminacy of power transformations of popular distributions such as lognormal, inverse Gaussian and logistic. The Stieltjes classes presented here are new. We also find numerical values of the index of dissimilarity for some particular cases.

Annals of Probability, 2000

The distribution of the integral of the absolute value of the Brownian bridge was expressed by Ci... more The distribution of the integral of the absolute value of the Brownian bridge was expressed by Cifarelli and independently by Johnson and Killeen in the form of a series. Rice obtained the corresponding probability density by numerical integration. Here we determine the exact tail asymptotics of this distribution, as well as the exact asymptotics of its density function for the large values of the argument.

The Annals of Probability, 2002

Applied Mathematics and Computation, 2005

Journal of Mathematical Analysis and Applications, 2005

Abstract. The density distribution function of the integral of the exponential Brownian motion is... more Abstract. The density distribution function of the integral of the exponential Brownian motion is determined explicitly in the form of a rapidly convergent series. 1.

The distribution function of the integral of the absolute value of the Brownian motion was expres... more The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takacs in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

arXiv: Probability, 2009

Eprint Arxiv 0904 1870, Apr 12, 2009

The Annals of Probability, 2000