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- Homework Statement
- Covariance function of a weakly stationary process.

- Relevant Equations
- .

On page 3 of the lecture notes for Stochastic Analysis, it says '##B(s,t)## is the covariance function ##\mathbb{E}[X_sX_t]-\mathbb{E}[X_s]\mathbb{E}[X_t]##. Then On page 5, it says the notes also say that 'the covariance function ##B(s,t)## of a strongly stationary stochastic process is invariant under time shifts so ##B(s,t)=f(s-t)## for some ##f##'. I'm wondering where did ##s-t## come from?? And I'm wondering how does it convey the information that the covariance function is invariant under time shifts, i.e., ##B(s,t)=B(s+h,t+h)##? This seemed to me like a non-sequitur, am I misunderstanding something obvious?

To top off my confusion, the next line says 'A stochastic process is weakly stationary, second-order stationary, or wide-sense stationary if ##B(s,t)=C(s-t)##' I'm wondering why call it ##C(s-t)## when it's already been called ##f(s-t)##? what information do the letters ##C## and ##f## imply here?? And then, it says ' ##C(t)=\mathbb{E}[X_{s+t}X_s]## for all ##s## is called the covariance function.' I'm wondering how ##C(t)## is related to ##C(s-t)##, and why the covariance function here has a different definition than the one provided earlier on page 3, although weakly stationary doesn't imply zero mean.

Thanks always for your help.

To top off my confusion, the next line says 'A stochastic process is weakly stationary, second-order stationary, or wide-sense stationary if ##B(s,t)=C(s-t)##' I'm wondering why call it ##C(s-t)## when it's already been called ##f(s-t)##? what information do the letters ##C## and ##f## imply here?? And then, it says ' ##C(t)=\mathbb{E}[X_{s+t}X_s]## for all ##s## is called the covariance function.' I'm wondering how ##C(t)## is related to ##C(s-t)##, and why the covariance function here has a different definition than the one provided earlier on page 3, although weakly stationary doesn't imply zero mean.

Thanks always for your help.

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