Vtyjkyuk

Let $(X_n)_n in mathbbN$ be a discrete-time markov chain with state space $S := 0,...,r-1$. Let $P$ be the corresponding transition matrix and $(p_0,...,p_r-1)$ the initial distribution vector. Let $K:= n_0 + ... + n_r-1$ and define $q(n_0,...,n_r-1)$ to be the probability that up to time $K$ state $0$ is visited exactly $n_0$ times, state $1$ is visited exactly $n_1$ times,..., state $r-1$ is visited exactly $n_r-1$. Now let

$$
F_K(x_0,...,x_r-1) := sum_n_0+...+n_r-1 = Kq(n_0,...,n_r-1)x_0^n_0...x_r-1^n_r-1
$$
be the generating function of the visits up to time $K$.

I have determine the following formula

$$
F_K(x_0,...,x_r-1) = (p_0x_0,...,p_r-1x_r-1)(PD)^K-1e
$$
where $D$ is the diagonal matrix with diagonal entries $x_0,...,x_r-1$ and $e = (1,...,1)^t in mathbbR^r$.

Now I want to compute

beginequation*
sum_K=1^inftyleft(sum_n_0 + ... + n_r-1 = Kq(n_0,...,n_r-1)x_0^n_0...x_r-1^n_r-1right) = sum_K=1^infty(p_0x_0,...,p_r-1x_r-1)(PD)^K-1e
endequation*

Here is my approach. Can someone tell me, whether it is true?

Let $x_0,...,x_r-1 in (-1,1)$. Then I have $||PD||_infty < 1$, where $||cdot||_infty$ is the maximum absolute row sum of the matrix, i.e. for a $m * n$ Matrix $A$

$$
||A||_infty := max_1leq i leq msum_j=1^n|a_i,j|
$$

Therefore I can apply the Neumann Series for matrices and I get

beginalign*
sum_K=1^inftyleft(sum_n_0 + ... + n_r-1 = Kq(n_0,...,n_r-1)x_0^n_0...x_r-1^n_r-1right) &= sum_K=1^infty(p_0x_0,...,p_r-1x_r-1)(PD)^K-1e\
&= (p_0x_0,...,p_r-1x_r-1) sum_K=0^inftyleft( PDright)^K e\
&= (p_0x_0,...,p_r-1x_r-1) left( E - PD right)^-1 e
endalign*

Is this approach correct?