Aspects of Randomization in Infinitely Divisible and Max-Infinitely Divisible Laws

Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated 01May2003) here we study certain aspects of randomization in infinitely divisible (ID) and max-infinitely divisible (MID) laws. They generalize ID and MID laws. In particular we study mixtures of ID & MID laws, its relation to random sums & random maximums, corresponding stationary processes & extremal processes and some of their properties. It is shown that mixtures of ID laws and mixtures of MID laws appear as limits of random sums and random maximums respectively. We identify a class of probability generating functions for N, the random sample size. A method to construct class-L laws is given.Comment: 10 page

Distributions of the same type: non-equivalence of definitions in the discrete case

Distributions of the same type can be discussed in terms of distribution functions as well as their integral transforms. For continuous distributions they are equivalent. In this note it is shown that it is not so in the discrete case.Comment: 3 pages (4 pages in the journal format

Another Look at Random Infinite Divisibility

The drawbacks in the formulations of random infinite divisibility in Sandhya (1991, 1996), Gnedenko and Korelev (1996), Klebanov and Rachev (1996), Bunge (1996) and Kozubowski and Panorska (1996) are pointed out. For any given Laplace transform, we conceive random (N) infinite divisibility w.r.t a class of probability generating functions derived from the Laplace transform itself. This formulation overcomes the said drawbacks, and the class of probability generating functions is useful in transfer theorems for sums and maximums in general. Generalizing the concepts of attraction (and partial attraction) in the classical and the geometric summation setup to our formulation we show that the domains of attraction (and partial attraction)in all these setups are same. We also establish a necessary and sufficient condition for the convergence to infinitely divisible laws from that of an N-sum and conversely, that is an analogue of Theorem.4.6.5 in Gnedenko and Korelev (1996, p.149). The role of the divisibiltiy of N and the Laplace transform on that of this formulation is also discussed.Comment: Added the Journal publication reference, in the journal format, 22 pages and typos are correcte

Maxwell's hypothesis reconsidered

Maxwell's derivaion of the distributions of the velocities of molecules is based on the assumption that the velocity components in the three mutualy orthogonal directions are independent. Here we note that his assumption, the phase space is isotropic, in fact nullifies the effect of a variety of dependencies among the velocity componenets. Thus we can do away with the independence assumption. Further, we observe that his conclusion regarding distribution of the velocity components (Gaussian) remains true under a set of weaker assumptions.Comment: 4 figure

A Max-AR(1) Model with Max-Semistable Marginals

The structure of stationary first order max-autoregressive schemes with max-semi-stable marginals is studied. A connection between semi-selfsimilar extremal processes and this max-autoregressive scheme is discussed resulting in their characterizations. Corresponding cases of max-stable and selfsimilar extremal processes are also discussed.Comment: In journal format, 5 Pages, contents change

Infinite Divisibility and Max-Infinite Divisibility with Random Sample Size

Continuing the study reported in Satheesh (2001),(math.PR/0304499 dated 01 May 2003) and Satheesh (2002)(math.PR/0305030 dated 02May 2003), here we study generalizations of infinitely divisible (ID) and max-infinitely divisible (MID) laws. We show that these generalizations appear as limits of random sums and random maximums respectively. For the random sample size N, we identify a class of probability generating functions. Necessary and sufficient conditions that implies the convergence to an ID (MID) law by the convergence to these generalizations and vise versa are given. The results generalize those on ID and random ID laws studied previously in Satheesh (2001b, 2002) and those on geometric MID laws studies in Rachev and Resnick (1991). We discuss attraction and partial attraction in this generalization of ID and MID laws.Comment: 14 pages, in journal format. In the first sentence of the last paragraph on page 131 the part after the second comma was inadvertently omitted and was missed even in the proof reading. This has been correcte

An Autoregressive Model with Semi-stable Marginals

The family of semi-stable laws is shown to be semi-selfdecomposable. Thus they qualify to model stationary first order autoregressive schemes. A connection between these autoregressive schemes with semi-stable marginals and semi-selfsimilar processes is given.Comment: PDF File, 5 Pages, corrections incorporated and contents change

A generalization of random self-decomposability

The notion of random self-decomposability is generalized here. Its relation to self-decomposability, Harris infinite divisibility and its connection with a stationary first order generalized autoregressive model are presented. The notion is then extended to $\mathbf{Z_+}$-valued distributions.Comment: 7 page

Medical Image Denoising using Adaptive Threshold Based on Contourlet Transform

Image denoising has become an essential exercise in medical imaging especially the Magnetic Resonance Imaging (MRI). This paper proposes a medical image denoising algorithm using contourlet transform. Numerical results show that the proposed algorithm can obtained higher peak signal to noise ratio (PSNR) than wavelet based denoising algorithms using MR Images in the presence of AWGN.Comment: 7 pages, 6 figures,Advanced Computing: An International Journal (ACIJ) ISSN: 2229 - 6727 [Online]; 2229 - 726X [Print

On the Marginal Distributions of Stationary AR(1) Sequences

In this note we correct an omission in our paper (Satheesh and Sandhya, 2005) in defining semi-selfdecomposable laws and also show with examples that the marginal distributions of a stationary AR(1) process need not even be infinitely divisible.Comment: 4 pages, in .pdf format, submitte