Consider the two recursions $$\theta_{n+1} = \theta_n + a(n)[a \theta_n + b w_n]$$ and
$$w_{n+1} = w_n + b(n)[c \theta_n + d w_n]$$
It is given that $\sum a(n) = \infty, \sum b(n) = \infty, \sum a(n)^2 < \infty, \sum b(n)^2 < \infty, \frac{a(n)}{b(n)} \to 0$. Let $\alpha_n = (\theta_n,w_n)$. Clearly, $\alpha_n = \Pi_{i=1}^{n-1}(I + M_i)\alpha_0$ where $M_i = \begin{bmatrix} \alpha a(i) & \beta a(i) \\ \gamma b(i) & \delta b(i) \\ \end{bmatrix}$. I want to know under what conditions on $\alpha,\beta,\gamma,\delta, a(n),b(n)$, $\|\alpha_n\|$ is uniformly bounded for all $n$ ?