Consider a vector stochastic process $e_{n}$ with process dynamics as $$e_{n+1}=(A_{n}\cdot e_{n})+r_{n}+s_{n}.$$ Here, $s_{n}$ is an indepedent, zero-mean Gaussian process; the spectral norm of the matrix $A_{n}$ is bounded by a real constant $a$ for all n; the term $r_{n}$ is such that $||r_{n}|| \le b\cdot||e_{n}||^{2}$ for $||e_{n}|| \le \epsilon$, where $||\cdot||$ denotes the $l_2$ norm, and $b$ is some real constant.
I want to know the upper bound on $||e_{n+1}||$, provided that $||e_{n}||\le \epsilon$?
I proceeded as follows. Using the properties of norms, we get $$||e_{n+1}|| \le (a*||e_{n}||)+||r_{n}||+||s_{n}||.$$ With $||e_{n}|| \le \epsilon$, this leads to $$||e_{n+1}|| \le (a \cdot \epsilon)+(b \cdot \epsilon^{2})+||s_{n}||.$$
However, it is not clear how to upper bound the norm of the Gaussian term $||s_{n}||$.
I came across this while checking the proof of Theorem 3.1 on Pg. 718, equation (79) in this paper.
Any help in this regard is highly appreciated.