Yes, but you can get other quantities with this approach as well (though not in nearly as nice a form). To do this, consider the random matrix

t = Bernoulli(T) (i.e., T = E[t])

The distribution is

m_n = M_1 t_1 ... t_n,

which has expected value

M_n = E[m_n] = E[M_1 t_1 ... t_n] = M_1 T^n

Here, each of the t_i is an i.i.d. sample of t. You can similarly compute the variance:

Var[m_n] = Var[M_1 t_1 ... t_n]

I don't know a closed form for this off the top of my head, but at this point simulations are probably the best way to go.