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Statistical Estimation of the Jump Activity for Time-changed Levy Processes

This paper is devoted to studying the problem of the statistical inference on the activity of jumps for a class of the so-called time-changed Levy processes, i.e., for the processes in the form Ys = XT (s), where X is a Levy process and T is a non-negative and non-decreasing stochastic process, which is referred to as time change. First, starting from some natural assumptions on the Levy measure of X, we infer on the asymptotic behavior of the characteristic function of Y. Next, we present a new method, which allows to consistently estimate the activity of small jumps in the dicult case of lowfrequency data.