LÚvy Processes and Infinitely Divisible Distributions
LÚvy processes are rich mathematical objects and constitute perhaps the most basic class of stochastic processes with a continuous time parameter. This book provides the reader with comprehensive basic knowledge of LÚvy processes, and at the same time introduces stochastic processes in general. No specialist knowledge is assumed and proofs and exercises are given in detail. The author systematically studies stable and semi-stable processes and emphasizes the correspondence between LÚvy processes and infinitely divisible distributions. All serious students of random phenomena will benefit from this volume.
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a-semi-stable a-stable absolutely continuous additive process assertion Assume Borel set bounded Brownian motion called characteristic function choose compound Poisson process condition continuous function continuous singular convergence Corollary defined Definition denoted density drift equivalent Example follows G Rd Gaussian Hence identical in law implies independent infinitely divisible distribution integral interval jumps Lebesgue measure Lemma Let Xt Levy measure Levy process Xt limsup linear linear subspace Markov process Markov property measurable function non-trivial nonnegative oo a.s. P[Xt parameter positive probability measure probability space process in law process on Rd proof of Theorem Proposition random variables random walk recurrent Remark respectively right-continuous right-hand side sample functions satisfying selfdecomposable distribution selfsimilar semi-selfsimilar semi-stable semigroup sequence stable processes stochastic process strictly stable strongly unimodal subordinator Suppose that Xt symmetric transient transition function triplet unimodal with mode uniquely