Stationary and Periodic Solutions of Differential Equations
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چکیده
A stochastic process ξ(t) = ξ(t,ω) (−∞ < t < ∞) with values in R is said to be stationary (in the strict sense) if for every finite sequence of numbers t1, . . . , tn the joint distribution of the random variables ξ(t1 + h), . . . , ξ(tn + h) is independent of h. If we replace the arbitrary number h by a multiple of a fixed number θ , h= kθ (k =±1,±2, . . . ), we get the definition of a periodic stochastic process with period θ , or a θ -periodic stochastic process.1 Stationary and periodic stochastic processes constitute a mathematical idealization of physical noise acting on linear and nonlinear devices functioning in a medium with unvarying or periodically varying properties. Let ξ(t) be a stationary stochastic process with finite variance. By the definition of stationarity,
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