Let $A$ be a discrete time Markov chain, with states $1$ and $2$, and transition probability matrix $$\begin{pmatrix} \frac{1}{2}( 1 + \alpha ) & \frac{1}{2}( 1 - \alpha ) \\ \frac{1}{2}( 1 - \alpha ) & \frac{1}{2}( 1 + \alpha ) \end{pmatrix} $$ where $-1 < \alpha < 1$.
Let $X(1)=R^{(1)}$ and $X(2)=R^{(2)}$, for real constants $R^{(1)}$ and $R^{(2)}$, satisfying $0 < R^{(1)} < R^{(2)} < \infty$. Let $\rho$ be the largest Perron-Frobenius eigenvalue of the matrix $$\begin{pmatrix} \frac{1}{2}( 1 + \alpha ) R^{(1)} & \frac{1}{2}( 1 - \alpha ) R^{(2)} \\ \frac{1}{2}( 1 - \alpha ) R^{(1)} & \frac{1}{2}( 1 + \alpha ) R^{(2)} \end{pmatrix}. $$
Let $n$ be an integer taking the values $n \in \lbrace -2, -1, 1, 2 \rbrace$.
Let $\mathbb{E}$ denote expectation. Define, for $n \in \lbrace -2, -1, 1, 2 \rbrace$,
$$G^{(\infty)}(n) \, = \, \lim_{K \rightarrow \infty} \mathbb{E}\left[ \prod_{j=1}^K \big( (X(A_{j})/\rho)^n \big) \right].$$
I need to show that there exist $\overline{L}(n)$ and $\overline{U}(n)$, satisfying $0 < \overline{L}(n)$ and $\overline{U}(n) < \infty$, such that $$\overline{L}(n) \, \leq \, G^{(\infty)}(n) \, \leq \, \overline{U}(n),$$ for all $n \in \lbrace -2, -1, 1, 2 \rbrace$. Here, $\overline{L}(n)$ and $\overline{U}(n)$ can be functions of the parameters $\alpha$, $R^{(1)}$, $R^{(2)}$, and $\rho$.
I am stuck.
I believe (in the special case that $n=1$) Horn and Johnson [2013, Theorem 8.2.7] implies $$0 \, < \, \underline{c} \, \leq G^{(\infty)}(1) \, \leq \, \overline{c} \, < \, \infty,$$ for constants $\underline{c}$ and $\overline{c}$, that are related to the eigenvectors of $A$. If correct, this would handle $n=1$.
Lemma: Let $V$ be a strictly positive random variable, with $\mathbb{E}[V] < \infty$. Then (using Cauchy-Schwarz and Jensen) $\mathbb{E}[V^2] \geq (\mathbb{E}[V])^2$ and $\mathbb{E}[1/V^2] \geq \mathbb{E}[1/V] \geq \frac{1}{\mathbb{E}[V]}$. $ \, $ $\blacksquare$
By the Lemma, and taking limits, I have a strictly positive lower bound for $G^{(\infty)}(n)$, for all $n \in \lbrace -2, -1, 1, 2 \rbrace$. However, I am stuck on the upper bound.
Questions:
How can I get a finite upper bound for $G^{(\infty)}(n)$, for all $n \in \lbrace -2, -1, 1, 2 \rbrace$?
Is there a ``clever" way to do all of this? My way above smacks of "brute force"?