Please help me understand a definition of a covariance function

[docnet]

Homework Statement

Covariance function of a weakly stationary process.

Relevant Equations

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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.

 

[FactChecker]

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., $B(s,t)=B(s+h,t+h)$?

Saying that $B(s,t)$ only depends on $s-t$ means that $B(s,t) = B(s+h, t+h)$ because $(s+h)-(t+h) =s-t$.

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 $f(s-t)$?

It's just a question of how broadly they want to make the definition of "covariance function". The function $f(s-t)$ was just introduced to say that a strongly stationary $B(s,t)$ only depends on $s-t$. $f(s-t)$ has not been defined as the covariance function for the strongly stationary process yet because they want to define the covariance function, $C(s-t)$, in the broader context of weakly stationary processes. Once "weakly stationary" and "covariance function" are defined, it is clear that "strongly stationary" implies "weakly stationary" and that $f(s-t)$ is the covariance function $C(s-t)$.

 

[WWGD]

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.

Isn't that a property of/definition of Brownian motion? Isn't that a property of the objects you're working with?