Two Gaussian processes with same variances and means but different covariances.

Let $W_t$ be standard Wiener process. Define $X_t = W_t$ and

$$ Y_t = \begin{cases} 0 & \text{when} & t = 0 \\ \displaystyle \frac {W_{t^2}} {\sqrt{t}} & \text{when} & t > 0 \end{cases} $$

Then both $X_t$ and $Y_t$ are Gaussian process, with common mean $$ E[X_t] = E[Y_t] = 0 $$

and common variance

$$ Var[X_t] = t = \frac {t^2} {(\sqrt{t})^2} = Var\left[\frac {W_{t^2}} {\sqrt{t}} \right] = Var[Y_t] $$

However the autocovariance is different:

$$ Cov[X_s, X_t] = \min\{s, t\} $$

$$ Cov[Y_s, Y_t] = \frac {\min\{s^2, t^2\}} {\sqrt{st}} = \begin{cases} \displaystyle \frac {s^2} {\sqrt{st}} = s \sqrt{\frac {s} {t}} \leq s & \text{when} & s \leq t \\ \displaystyle \frac {t^2} {\sqrt{st}} = t \sqrt{\frac {t} {s}} < t & \text{when} & s > t \end{cases}$$

So we have $$ Cov[Y_s, Y_t] \leq Cov[X_s, X_t] $$ with equality holds only when $s = t$.