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S. Satheesh

Publications and source records attributed to S. Satheesh.

17 recordsLinked to original sources
arXiv2017

On Certain Colouring Parameters of Mycielski Graphs of Some Graphs

Colouring the vertices of a graph $G$ according to certain conditions can be considered as a random experiment and a discrete random variable $X$ can be defined as the number of vertices having a particular colour in the proper colouring of $G$. The concepts of mean and variance, two important statistical measures, have also been introduced to the theory of graph colouring and determined the values of these parameters for a number of standard graphs. In this paper, we discuss the colouring parameters of the Mycielskian of certain standard graphs.

math.GM↗
arXiv2017

On Certain Colouring Parameters of Graphs

Colouring the vertices of a graph $G$ according to certain conditions can be considered as a random experiment and a discrete random variable $X$ can be defined as the number of vertices having a particular colour in the proper colouring of $G$. In this paper, we extend the concepts of mean and variance, two important statistical measures, to the theory of graph colouring and determine the values of these parameters for a number of standard graphs.

math.GM↗
arXiv2014

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.

physics.gen-ph↗
arXiv2014

N-max infinite divisibility and N-max stability

Here we give a necessary and sufficient condition for the convergence to a random max infinitely divisible law from that of a random maximum. We then discuss random max-stable laws, their domain of max-attraction and the associated extremal processes.

math.PR↗
arXiv2011

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.

cs.OH↗
arXiv2008

Geometric Gamma Max-Infinitely Divisible Models

A transformation of gamma max-infinitely divisible laws viz. geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal process corresponding to this distribution is presented. A new kind of invariance (stability) under geometric maxima is proved and a max-AR(1) model corresponding to it is also discussed.

math.PR↗
arXiv2005

On the Stability of Geometric Extremes

Possible reasons for the uniqueness of the positive geometric law in the context of stability of random extremes are explored here culminating in a conjecture characterizing the geometric law. Our reasoning comes closer in justifying the geometric law in similar contexts discussed in Arnold et al. (1986) and Marshall & Olkin (1997) and also supplement their arguments.

math.PR↗
arXiv2005

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.

math.PR↗
arXiv2004

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.

math.PR↗
arXiv2004

Why There Are No Gaps In The Support Of Non-Negative Integer-Valued Infinitely Divisible Laws?

Remark.9 in Bose-Dasgupta-Rubin (2002) review states that when a non-negative integer-valued infinitely divisible law has an atom at unity then its support cannot have any gaps. Here one has two questions. (i) Why there are no gaps and (ii) Can there be gaps if the condition is not satisfied. Our investigation with these questions in mind centers on the implications of having and not having atoms at zero and unity. We give two examples/ constructions, which show that the remark needs modification and we modify it.

math.PR↗
arXiv2003

Stability of Random Sums

When the distribution of a random (N) sum of independent copies of a r.v X is of the same type as that of X we say that X is N-sum stable. In this paper we consider a generalization of stability of geometric sums by studying distributions that are stable under summation w.r.t Harris law. We show that the notion of stability of random sums can be extended to include the case when X is discrete. Finally we propose a method to identify the probability law of N for which X is N-sum stable. See also Satheesh and Nair (2002), (Some classes of distributions on ther non-negative lattice, J. Ind. Statist. Assoc., 2002, 40, 41-58) for a study of discrete laws of the same type and stability of geometric sums of discrete laws.

math.PR↗
arXiv2003

Some Classes Of Distributions On The Non-Negative Lattice

A method for constructing distributions on the non negative integers as discrete analogue of continuous distributions on the non negative real is presented. A justification of the definition of discrete self decomposable laws is provided. Discrete analogue of distributions of the same type and the role of Bernoulli law in this context is discussed. Generalizations of some discrete laws and their properties are given. The geometric compounding problem for discrete distributions is studied by introducing discrete semi Mittag Leffler laws.

math.PR↗
arXiv2003

A Supplement To The Bose-Dasgupta-Rubin (2002) Review Of Infinitely Divisible Laws And Processes

This paper proves that if a discrete distribution is infinitely divisible (ID) with integer-valued components, then it has a mass at the origin, which also implies why certain ID discrete laws do not have gaps in its support. We argue that discrete laws also can be stable and such laws do have domain of attraction. Then we give certain recent developments and references not reported in the Bose Dasgupta Rubin (2002) review in Sankhya, and some examples in the topics; infinite divisibility and stability of discrete laws, random infinite divisibility, operator stable laws, class-L laws, Goldie-Steutel result, max-infinite divisibility and stability, simulation, alternate stable laws, applications and free probability theory.

math.PR↗
arXiv2003

A Note on Maximum and Minimum Stability of Certain Distributions

In the context of stability of the extremes of a random variable X with respect to a positive integer valued random variable N we discuss the cases (i) X is exponential (ii) non-geometric laws for N (iii) identifying N for the stability of a given X and (iv) extending the notion to a discrete random variable X.

math.PR↗
arXiv2003

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.

math.PR↗