The Lyapunov exponent can be used to describe the asymptotic growth rate of the norm of a product of random matrices acting on a specific vector $X$, and is defined as:
$$ \lambda = \lim_{n\to\infty}\sup \frac{1}{n}\log || A_n \cdots A_1 X||,$$
where $A_i$ are i.i.d. $d \times d$ matrices whose norms have finite expectation. Oseledet's multiplicative ergodic theorem states that the maximum number of distinct Lyapunov exponents is $d$.
My question is, can anyone help me find a general form for the $2$ Lyapunov exponents of a matrix:
$$A=\left[\begin{array}{cc} 0 & 1\\ a & b \end{array}\right], $$
where $a$ and $b$ are real random variables with $\mathbb{E}[\log a]<\infty$ and $\mathbb{E}[\log b]<\infty$?
Notoriously, Lyapunov exponents of random matrices are seldom explicit. You are trying to determine the behaviour of the random sequence $(x_n)$ defined by $x_{n+1}=a_nx_n+b_n$, where $(a_n,b_n)$ is i.i.d. For some informations about these systems, see the survey "Iterated random functions" by Diaconis and Freedman (1999).