The Kontorovich-Lebedev Transform

It is pointed out that difficulties can occur with the Kontorovich-Lebedevtransform when expressed in terms of Hankel functions and anew form is proposed which overcomes these difficulties.     

The Kontorovich-Lebedev integral transform

One considers the Kontorovich-Lebedev integral transform

The Kontorovich-Lebedev integral transformation with a Hankel function kernel in a space of generalized functions of doubly exponential descent

A version of the Kontorovich-Lebedev transformation with the Hankel function of second kind in the kernel is investigated in a space of distributions of doubly exponential descent. The inversion theorem is rigorously established making use in some steps of the proof of a relation of this transform with the Laplace one. Finally, the theory developed is illustrated in solving certain type of partial differential equations.     

Inversion of the Kontorovich-Lebedev integral transform

Translated from Matematicheskie Zametki, Vol. 51, No. 5, pp. 27–34, May, 1992.     

Extending circular distributions through transformation of argument

This paper considers the general application to symmetric circular densities of two forms of change of argument: one produces extended families of distributions which contain symmetric densities which are more flat-topped, as well as others which are more sharply peaked, than the originals, and the second produces families which are skew. General results for the modality and shape characteristics of the densities which ensue are presented, and maximum likelihood estimation of the parameters of two extensions of the Jones–Pewsey family is discussed. The application of these two particular extended families is illustrated within analyses of data on monthly cases of sudden infant death syndrome in the UK.     

Scaling transformation and probability distributions for financial time series

The price of financial assets are, since [Bachelier L. Annales de l'Ecole Normale Supérieure 1900;3:XVII:21–86], considered to be described by a (discrete or continuous) time sequence of random variables, i.e., a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [Mandelbrot BB. J Business 1963;36:394–419; Peters EE. Chaos and order in the capital markets. New York: Wiley, 1991; Mantegna RN, Stanley HE. Nature 1995;376:46–49; Evertsz CJG. Fractals. 1995;3:609–616; Bouchaud JP, Potters M. Théorie des risques financiers. Aléa Saclay, 1997]. In this paper we investigate the question of scaling transformation of price processes by establishing a new connection between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.     

An Index Integral and Convolution Operator Related to the Kontorovich-Lebedev and Mehler-Fock Transforms

We deal with an index integral involving the product of the modified Bessel functions and associated Legendre functions. It was discovered by Ferrell (Nucl Instrum Methods Phys Res B 96:483?C485, 1995) while comparing solutions of the Laplace equation in different coordinate systems in his study of the so-called surface plasmons in various condensed matter samples. This integral is quite interesting from the pure mathematical point of view and it is absent in famous reference books for series and integrals. We give a rigorous proof of this formula and discuss its particular cases. We also construct a convolution operator associated with this integral, which is related to the classical Kontorovich-Lebedev and Mehler-Fock transforms. Mapping properties and the norm estimates in weighted L _p -spaces, 1 ?? p ?? 2 are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

The asymptotic distributions of sums of record values for distributions with lognormal-type tails

Let X1,X2,... be a sequence of i.i.d. random variables and let X(1),X(2),... be the associatedrecord value sequence. We focus on the asymptotic distributions of sums of records, Tn=∑nk=1X(k), forX1 ∈ LN(γ). In this case, we find that 2 is a strange point for parameter γ. When γ＞ 2, Tn is asymptoticallynormal, while for 2 ＞γ＞ 1, we prove that Tn cannot converge in distribution to any non-degenerate lawthrough common centralizing and normalizing and log Tn is asymptotically normal.     

The canonical decomposition of bivariate distributions

The ordinary notion of a bivariate distribution has a natural generalisation. For this generalisation it is shown that a bivariate distribution can be characterised by a Hilbert space and a family _p, 0 ≤ p ≤ 1, of subspaces of . specifies the marginal distributions whilst _p is a summary of the dependence structure. This characterisation extends existing ideas on canonical correlation.     

The asymptotic distributions of incomplete U-statistics

Summary An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.     

The theory of concentrated Langevin distributions

The density of the Langevin (or Fisher-Von Mises) distribution is proportional to exp κμ′x, where x and the modal vector μ are unit vectors in $\text{R}$^q. κ (≥0) is called the concentration parameter. The distribution of statistics for testing hypotheses about the modal vectors of m distributions simplify greatly as the concentration parameters tend to infinity. The non-null distributions are obtained for statistics appropriate when κ₁,…,κ_m are known but tend to infinity, and are unknown but equal to κ which tends to infinity. The three null hypotheses are H₀₁:μ = μ₀(m=1), H₀₂:μ₁ = … =μ_m, H₀₃:μ_i?V, i=1,…,m In each case a sequence of alternatives is taken tending to the null hypothesis.     

The generalized hypergeometric family of distributions

Summary This is an expository summary of the authors' report on classification of the generalized hypergeometric (GHg for short) family of distributions (Sibuya and Shimizu (1981),Keio Science and Technology Report, to appear). Emphasis is laid on the definition of the distributions based on some conventional rules, and on the complete classification of the multivariate GHg distributions, whose types are found to be rather limited in spite of their quite general definition. Previous classifications and namings are summarized and compared with the new one.     

The convolution of functions and distributions

The non-commutative convolution f∗g of two distributions f and g in D^′ is defined to be the limit of the sequence {(fτn)∗g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that

     

The rate of convergence for subexponential distributions

A distribution functionF on the nonnegative real line is called subexponential if