Assume state $0$ is positive recurrent. We take the initial state to be $0$. Let $\{W_n\}$, $n=1,\ldots,n$ denote successive recurrence times which are of course independent and identically distributed random variables with finite mean and with a generating function $$ F(t)= \sum_{k=1}^{\infty} t^k \ Pr\{W_1=k\}, \ \ \ |t|<1$$
Define $Y_n$ as the time of the last visit to state $0$ before the time $n$. With this I have to prove that
$$\sum_{n=0}^{\infty} t^n \sum_{j=0}^{n} x^j Pr\{ Y_n=j\} = \frac{1-F(t)}{(1-t)(1-F(xt))} $$
For this I have to prove this $$Pr\{Y_n=j \}=Pr\{W_1+\cdots+W_{N_n}=j\} \cdot q_{n-j}$$ where $q_i=Pr\{W_1>i\}$ and $N_n$ is the number of visits to state $0$ in the first $n$ trials. (As a hint of my teacher)
Ok, first I tried to analyze $Pr\{Y_n=j \}$:
$$ Pr\{Y_n=j \}= Pr\{X_j=0, X_k \ne 0, j+1 \leq k \leq n \}$$ $$ = Pr\{ X_j=0, W_1 > n-j \} = Pr \{ W_1 > n-j | X_j=0\} Pr\{X_j=0\} $$ $$= Pr\{X_j=0 \} Pr\{W_1 > n-j \} = Pr\{X_j=0 \} q_{n-j} $$
and here I stuck, I think this $= q_{n-j} \sum_{k=0}^j Pr\{W_1+\cdots+W_k=j\}$ but i can not prove that $Pr\{Y_n=j \}=Pr\{W_1+\cdots+W_{N_n}=j\} \cdot q_{n-j}$
Is this correct? Could someone help me to conclude this problem, pls.
Please some help....
Thanks for your time and help.