Why random signals are called power signals?

Energy signals are finite duration. That means if a random signal is energy type, all its realizations or its sample functions must be zero at infinity. So its expected value or mean is zero at infinity.

On the other hand, a stationary process has a time invariant mean. So if its expected value is zero in infinity, it will be zero in any time.

In summary if a stationary random signal is energy, its expected value must be zero every time. But It is not correct for all signals. So it is not energy type.