Please help me understand a definition of a covariance function

docnet · Mar 5, 2024

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 · Mar 5, 2024

docnet said:

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##.

docnet said:

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 · Mar 5, 2024

docnet said:

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?

Please help me understand a definition of a covariance function

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