Traditionally a weak stationary process is also called covariance stationary, but those 3 properties are exposed:

$$E[Xt] = μ , \forall t$$ $$var(Xt) = \sigma^2, \forall t$$ $$cov(Xt, Xt−j) = \gamma_j, \forall t$$

that are respectively mean, variance and covariance stationarity. We can merge 2 and 3 by extending to $j=0$ and $\gamma_0 = \sigma^2$, and let's call it covariance-only stationarity

But seeing the name covariance stationarity and also trying to figure out examples where the variance is stationary and not the mean, or where covariances (j=0 too) are stationary but not the mean, I was wondering if maybe covariance-only stationarity could lead to mean stationarity.

1. Is there a proof covariance-only stationarity → mean stationarity?
2. or are both covariance-only stationarity and mean stationarity required for having a covariance stationary process?

note: if 2., in my opinion 'covariance stationary' is a very misleading term, weak stationary would be better to use

Traditionally a weak stationary process is also called covariance stationary, but those 3 properties are exposed:

$$E[Xt] = μ , \forall t$$ $$var(Xt) = \sigma^2, \forall t$$ $$cov(Xt, Xt−j) = \gamma_j, \forall t$$

that are respectively mean, variance and covariance stationarity. We can merge 2 and 3 by extending to $j=0$ and $\gamma_0 = \sigma^2$, and let's call it covariance-only stationarity

But seeing the name covariance stationarity and also trying to figure out examples where the variance is stationary and not the mean, or where covariances (j=0 too) are stationary but not the mean, I was wondering if maybe covariance-only stationarity could lead to mean stationarity.

1. Is there a proof for covariance-only stationarity → mean stationarity?
2. or are both covariance-only stationarity and mean stationarity required for having a covariance stationary process?

note: if 2., in my opinion 'covariance stationary' is a very misleading term, weak stationary would be better to use

Traditionally a weak stationary process is also called covariance stationary, but those 3 properties are exposed:

$$E[Xt] = μ , \forall t$$ $$var(Xt) = \sigma^2, \forall t$$ $$cov(Xt, Xt−j) = \gamma_j, \forall t$$

that are respectively mean, variance and covariance stationarity. We can merge 2 and 3 by extending to $j=0$ and $\gamma_0 = \sigma^2$

But seeing the name covariance stationarity and also trying to figure out examples where the variance is stationary and not the mean, or where covariances (j=0 too) are stationary but not the mean, I was wondering if maybe covariance stationarity in its extended form (j=0 too) could lead to mean stationarity.

Is there a proof for it, that would justify this name?

Traditionally a weak stationary process is also called covariance stationary, but those 3 properties are exposed:

$$E[Xt] = μ , \forall t$$ $$var(Xt) = \sigma^2, \forall t$$ $$cov(Xt, Xt−j) = \gamma_j, \forall t$$

that are respectively mean, variance and covariance stationarity. We can merge 2 and 3 by extending to $j=0$ and $\gamma_0 = \sigma^2$, and let's call it covariance-only stationarity

But seeing the name covariance stationarity and also trying to figure out examples where the variance is stationary and not the mean, or where covariances (j=0 too) are stationary but not the mean, I was wondering if maybe covariance-only stationarity could lead to mean stationarity.

1. Is there a proof covariance-only stationarity → mean stationarity?
2. or are both covariance-only stationarity and mean stationarity required for having a covariance stationary process?

note: if 2., in my opinion 'covariance stationary' is a very misleading term, weak stationary would be better to use