Applications of smoothly varying functions and tail index estimation
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This dissertation focuses on heavy tail analysis. The most important classes of heavy tail distributions are the class of regularly varying distribution functions and the class of subexponential distributions. The class of smoothly varying functions, i.e. functions with continuous derivatives being regularly varying at in¯nity, is a subclass of regularly varying functions. Under an assumption of smooth variation on the step size distribution we obtain higher order expansions for the tail distribution of the global maximum of random walk with negative drift, and under mild regularity conditions that of randomly stopped sums of subexponential random variables with smoothly varying hazard function. Tail index is the key parameter for distributions with regularly varying tail. We study the asymptotic properties of Hill's estimator, one of the commonly used tail index estimators, for shot noise sequences and ¯rst-order bifurcating autoregressive processes.