Let Y = c(n)Xn, where Xn is a zero-mean, unit-variance, discrete-time random process and c(n) is a deterministic function of n. (a) Find the mean and variance of Y. (b) Find the joint cdf of Yn and Yn 1. (c) Find the autocovariance function of Yn.
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SOLVED:Solve Eq. (2.4) for the continuous-time random walk by first Laplace-transforming to give the difference equation s P(n, s)-P(n, t=0)=P(n+1, s)+P(n-1)-2 P(s) for the Laplace transform P(n, s)=∫0^∞ Pn(t) e^-s t d t . Show that P(n, s)= λ-^n /(s+1-λ-), where λ±=(1 ±√(1-4 a^2)) / 2 a and a=(s+2)^-1. Show that in the long-time limit, corresponding to s →0 in the Laplace transform, inversion of the Laplace transform recovers the Gaussian distribution (2.6) for Pn(t).
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