The time series $\{X_t\}$ is said to be weak stationary if for all $n > ≥ 1$ for any $t_1, . . . ,t_n ∈ T$ and for all $τ$ such that $t_1+\tau, \dots ,t_n + τ ∈ T$ all the joint moments of order 1 and 2 of $X_{t_1} , . . . , X_{t_n}$ exist, are all finite and equal to the corresponding joint moments of $X_{t_1+\tau} , . . . , X_{t_n+\tau}$ . In fact this corresponds to $\Bbb E\{X_t\} = µ, Var\{X_t\} = σ^2$ and $\Bbb E\{X_{t_1}X_{t_2} \} = \Bbb E\{X_{t_1+\tau}X_{t_2+\tau} \}$. One may deduce from this that $\Bbb E\{X_{t_1}X_{t_2} \}$ is a function of $t_2 − t_1$ only.
I do not understand the last statements of the above. First, I never heard about joint moments, is is just $\Bbb E\{( X_{t_i}\dots X_{t_n})^k\}$ ? If yes, then why is it said that $\Bbb E\{X_{t_1}X_{t_2} \} = \Bbb E\{X_{t_1+\tau}X_{t_2+\tau} \}$ and not $\Bbb E\{X_{t_1}\dots X_{t_n} \} = \Bbb E\{X_{t_1+\tau}\dots X_{t_n+\tau} \}$ ? (i.e. why do they stop at $t_2$ ?). Finally, I do not understand how we deduce from this that $\Bbb E\{X_{t_1}X_{t_2} \}$ is a function of $t_2 − t_1$ only, since $X_i$ can depend on a lot of things, if $X_i$s are $Ber(p)$ random variables, then I do not see why would the quantity $\Bbb E\{X_{t_1}X_{t_2} \}$ not depend on $p$ and depend just on $t_2 − t_1$. Can someone explain ?