Applied Stochastic Processes Problem Set 8

Recall, from Theorem 8.1-2 on page 490 in [4], that a random process X(t) with autocorrelation function RXX(t1, t2) has a m.s. derivative at time t if ∂RXX(t1, t2)/∂t1∂t2 exists at t1 = t2 = t. Furthermore, we recall that the correlation function RXX(t1, t2) is related to the covariance function KXX(t1, t2) as follows. RXX(t1, t2) = KXX(t1, t2) + μX(t1)μX(t2) (1) For this problem we have a constant mean function μX 6= 0 such that RXX(t1, t2) = σ cos (ω0(t1 − t2)) + μXμX . Evaluating the mixed partial derivative of RXX(t1, t2) at t1 = t2 = t yields the following result. ∂RXX(t1, t2) ∂t1∂t2 ∣∣∣∣ (t1,t2)=(t,t) = σ { ∂2 cos (ω0(t1 − t2)) ∂t1∂t2 ∣∣∣∣ (t1,t2)=(t,t) } = σω 0