What is a covariance stationary process?
Ans. A covariance stationary process or sequence is a sequence is random variables having the same mean and the covariance between the any two terms of the sequence depends on the relative positions and not on the absolute positions of the two terms.
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Is the following MA(2) process covariance-stationary? Yt = (1 + 2.4L + 0.8L2) εt, with E ( εt εT) = $$ \left\{\begin{array}{c} 1 \text { for } t=\tau \\ 0 \text { otherwise } \end{array}\right. $$ If so, calculate its auto-covariances.
b) In what follows, we assume cc return r_t is a covariance stationary process. Prove the following statements: i. If r_t iid(0; sigma^2) (or independent white noise); then r_t mds(0; sigma^2). ii. If r_t mds(0; sigma^2); then r_t WN(0; sigma^2) (or weak white noise).
Let X(t) and Y(t) be independent, wide-sense stationary random process with zero means and the same covariance function Cx(t) Let Z(t) be defined by Z(t) = X(t)coswt + Y(t)sinwt Find the joint pdf of X(t1) and X(t2) in part b
2. Consider a following time series process Yt = 1.5Yt−1 −0.5Yt−2 +εt a) Rewrite this process in lag polynomial form. b) Is this process invertible? Is this process covariance stationary? c) Difference this process once and show that ΔYt = Yt −Yt−1 is covariance stationary.
a) Consider the following moving average process, MA(2): Yt = ut + α1ut-1 + α2ut-2 where ut is a white noise process, with E(ut)=0, var(ut)=σ2 and cov(ut,us)=0 . Derive the mean, E(Yt), the variance, var(Yt), and the covariances cov( Yt,Yt+1 ) and cov(Yt,Yt+2 ), of this process. b) Give a definition of a (covariance) stationary time series process. Is the MA(2) process (covariance) stationary?
What is meant by a stationary random process? Describe an example of a nonstationary random signal, and sketch 2 or 3 example sample functions for your nonstationary random process/signal.
3. Brounian motion f(O)eR+ is a special case of a Gaussian process with mean zero and covariance C(s, t) = min(s, t) (a) What is the distribution of f(1), the Brownian motion at time t = 4? (Hint: it may be useful to function recall that for any random variable X, var(X)-(x, X) (b) Fix tE R. What is the distribution of f(t)? (c) What is the distribution of f(4)-f(2)? (Hint: it may be useful to utilize var(X-Y) = var(X)...
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain. onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
Let { be a zero-mean stationary process and let a and b be constants. (a) (5 points) If Xi a+bt+St+Yi, where St is a seasonal component with period 12, show that ▽12V is stationary and express its autocovariance function in terms of that of { (b) (5 points) If X1-(a + bt)Sİ + Y. where Sı is a seasonal component with period 12, show that Vi2 is stationary and express its autocovariance function in terms of that of {
1. Let {Xt} be a stationary process with mean μt = E(Xt) = 0 and autocovariance function γX(k) = E(XtXt−k) - μ2 = E(XtXt+k) - μ2. De ne Yt = 5 + 2t + Xt. (a) Find E(Yt), the mean function for Yt. (b) Find γY (k), the autocovariance function for Yt in terms of γX (k). (c) Is Yt stationary? Explain. (d) De ne a new process Wt as Wt = Yt − Yt−1. Find E(Wt) and γW (k)....