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2S0ME FUNDAMENTAL CONCEPTS AND RESULTS OF MATHEMATICAL
FOUNDATIONS OF THE THEORY OF STOCHASTIC PROCESSES
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a-field applications assume asymptotic Borel sets Chapter complex-valued concerned conditional probability consider constant continuous sample functions corresponding covariance function covariance matrix defined definition denote derivative dF(X discussion distribution function downcrossings elementary events equivalent ergodic example exists fact finite-dimensional distributions fixed follows formula frequency further given Hence Hilbert space independent inequality input lemma limit linear mean number nonnegative nonstationary normal stationary process notation number of crossings obtain orthogonal increments parameter particular Poisson process probability space process f process f(f proof properties proved q.m. integral quadratic mean random variables real-valued relation result rj(t sample functions satisfied sequence spectral density spectral representation spectrum stationary normal process stationary stream stochastic process stream of events strictly stationary process subspace sufficient condition Suppose tangencies theorem theory upcrossings variable f variance vector waveform Wiener process write zero mean