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Extreme value theory for continuous-time (and discrete-time) stochastic processes has meanwhile a long history.
More general shot noise processes can be defined by replacing the Poisson process by a point process, e. g. a renewal process, and {Zjf (t)}t≥0 by {Xj (t)}t≥0, j ∈ N, an i. i. d. sequence of stochastic processes.
Then we call a stochastic process Y = {Yn}n∈Z with
The mark is the stochastic process Y under the conditional probability that Y (α) > uT .
Let Y = {Y (t)}t∈R be a stochastic process in R, which is a. s. bounded on every compact set and has the decomposition
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