Given that $X_n$ given $X_{n-1},...,X_0$ is Poisson distributed with mean $a+bX_{n-1}+cX_{n-2}$, for $n\geq 2, a>0,b,c\geq 0$. Define $Y_n=\begin{pmatrix}X_n\\X_{n-1} \end{pmatrix}$. Prove that $Y=(Y_n)_{n\in\mathrm{N}}$ is recurrent if $b+c<1$.
I know that if $X_{n}$ given $X_{n-1},...,X_0$ is Poisson distributed with mean $\lambda=a+bX_{n-1}$, where $a>0,b\geq 0$, then $X=(X_n)_{n\in\mathrm{N_0}}$ is a recurrent M.C & it is recurrent if $b <1$. Can we make use of this fact and extend it to the case of 2nd-order MC as shown above?
Hint: Prove that when $b+c\lt1$ there exists $\lambda\lt1$, $w\gt0$ and a vector $V$ with positive coordinates such that $Z=V^*Y-w$ solves $\mathbb E[Z_{n+1}\mid \mathcal F_n]=\lambda Z_n$.