Why does it mean that the process/signal is not stationary when its variance varied with time? that is,
$VAR[X(t)]= \alpha \times t$,$t$ is time,and $\alpha$ is a constant,then $X(t)$ is not the WSS process
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From the Wiki page - A stationary random process is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance, if they are present, also do not change over time.
The implication from the definition is that the mean and variance of random process do not vary over time. Since by your definition of the variance - it varies over time, then your process is non-stationary.
A couple of notes: