Stability of Random Sums and Extremes

This study is about the stability of random sums and extremes.The difficulty in finding exact sampling distributions resulted in considerable problems of computing probabilities concerning the sums that involve a large number of terms.Functions of sample observations that are natural interest other than the sum,are the extremes,that is , the minimum and the maximum of the observations.Extreme value distributions also arise in problems like the study of size effect on material strengths,the reliability of parallel and series systems made up of large number of components,record values and assessing the levels of air pollution.It may be noticed that the theories of sums and extremes are mutually connected.For instance,in the search for asymptotic normality of sums ,it is assumed that at least the variance of the population is finite.In such cases the contributions of the extremes to the sum of independent and identically distributed(i.i.d) r.vs is negligible.

Modeling and Analysis of Some Heavy Tailed Time Series

The thesis has covered various aspects of modeling and analysis of finite mean time series with symmetric stable distributed innovations. Time series analysis based on Box and Jenkins methods are the most popular approaches where the models are linear and errors are Gaussian. We highlighted the limitations of classical time series analysis tools and explored some generalized tools and organized the approach parallel to the classical set up. In the present thesis we mainly studied the estimation and prediction of signal plus noise model. Here we assumed the signal and noise follow some models with symmetric stable innovations.We start the thesis with some motivating examples and application areas of alpha stable time series models. Classical time series analysis and corresponding theories based on finite variance models are extensively discussed in second chapter. We also surveyed the existing theories and methods correspond to infinite variance models in the same chapter. We present a linear filtering method for computing the filter weights assigned to the observation for estimating unobserved signal under general noisy environment in third chapter. Here we consider both the signal and the noise as stationary processes with infinite variance innovations. We derived semi infinite, double infinite and asymmetric signal extraction filters based on minimum dispersion criteria. Finite length filters based on Kalman-Levy filters are developed and identified the pattern of the filter weights. Simulation studies show that the proposed methods are competent enough in signal extraction for processes with infinite variance.Parameter estimation of autoregressive signals observed in a symmetric stable noise environment is discussed in fourth chapter. Here we used higher order Yule-Walker type estimation using auto-covariation function and exemplify the methods by simulation and application to Sea surface temperature data. We increased the number of Yule-Walker equations and proposed a ordinary least square estimate to the autoregressive parameters. Singularity problem of the auto-covariation matrix is addressed and derived a modified version of the Generalized Yule-Walker method using singular value decomposition.In fifth chapter of the thesis we introduced partial covariation function as a tool for stable time series analysis where covariance or partial covariance is ill defined. Asymptotic results of the partial auto-covariation is studied and its application in model identification of stable auto-regressive models are discussed. We generalize the Durbin-Levinson algorithm to include infinite variance models in terms of partial auto-covariation function and introduce a new information criteria for consistent order estimation of stable autoregressive model.In chapter six we explore the application of the techniques discussed in the previous chapter in signal processing. Frequency estimation of sinusoidal signal observed in symmetric stable noisy environment is discussed in this context. Here we introduced a parametric spectrum analysis and frequency estimate using power transfer function. Estimate of the power transfer function is obtained using the modified generalized Yule-Walker approach. Another important problem in statistical signal processing is to identify the number of sinusoidal components in an observed signal. We used a modified version of the proposed information criteria for this purpose.

Median Sets and Median Number of a Graph

A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances to the vertices of the profile is the median of the profile. Any subset of the vertex set such that it is the median of some profile is called a median set. The number of median sets of a graph is defined to be the median number of the graph. In this paper, we identify the median sets of various classes of graphs such as Kp − e, Kp,q forP > 2, and wheel graph and so forth. The median numbers of these graphs and hypercubes are found out, and an upper bound for the median number of even cycles is established.We also express the median number of a product graph in terms of the median number of their factors.

Median graphs, the remoteness function, periphery transversals, and geodetic number two

A periphery transversal of a median graph G is introduced as a set of vertices that meets all the peripheral subgraphs of G. Using this concept, median graphs with geodetic number 2 are characterized in two ways. They are precisely the median graphs that contain a periphery transversal of order 2 as well as the median graphs for which there exists a profile such that the remoteness function is constant on G. Moreover, an algorithm is presented that decides in O(mlog n) time whether a given graph G with n vertices and m edges is a median graph with geodetic number 2. Several additional structural properties of the remoteness function on hypercubes and median graphs are obtained and some problems listed