LÚvy Processes and Infinitely Divisible Distributions
Lvy processes are rich mathematical objects and constitute perhaps the most basic class of stochastic processes with a continuous time parameter. This book is intended to provide the reader with comprehensive basic knowledge of Lvy processes, and at the same time serve as an introduction to stochastic processes in general. No specialist knowledge is assumed and proofs are given in detail. Systematic study is made of stable and semi-stable processes, and the author gives special emphasis to the correspondence between Lvy processes and infinitely divisible distributions. All serious students of random phenomena will find that this book has much to offer.
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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