Theorem

random sequences are sequences of random variables

$\mathbf{X} = (X_1 , X_2 , \dots )$ is a random sequence if and only if for each $i = 1, 2, \dots$ it is the case that $X_i$ is a random variable. Furthermore $\mathbf{X}$ is a random sequence if and only if for each $n$ the tuple $(X_1, \dots , X_n )$ is a random vector.