Probably a very simple question, I think I'm looking for a general statement that says 'if one decreases the independence between two processes then the expected value of the maximum of these two processes decreases'. 


More precisely, let $A\_0, A\_1$ be two non-negative $2\times 2$ matrices. Let $n\geq 6$ and $p\in(0,1)$. I want to compare


(1) $\mathbb E (\max\{||A\_{i\_1}\cdots A\_{i\_n}||, ||A\_{j\_1}\cdots A\_{j\_n}||\}:$ each $i\_k, j\_k\in\{0,1\}$ is picked independently with probability $(p,1-p))$ 


(2) $\mathbb E (\max\{||A\_{i\_1}\cdots A\_{i\_n}||, ||A\_{j\_1}\cdots A\_{j\_n}||\}:$ each $i\_k, j\_k\in\{0,1\}$ is picked independently with probability $(p,1-p)$ apart from $j\_5$, which is fixed at $j\_5=i\_3)$.


The $i\_3=j\_5$ condition is just meant to be some kind of dependence condition that isn't easy to factor out (I don't want to be able to write $A\_{i\_1}\cdots A\_{i\_n}$ and $A\_{j\_1}\cdots A\_{j\_n}$ as a product of 'the dependent bit' and 'the independent bit').


I would like to say that the value of (1) is greater than or equal to the value of (2), and that this follows from some general principle in probability theory that introducing any kind of dependence between two random processes decreases the expected value of the maximum of these two processes. Unfortunately I don't know my way around the probability literature, so any tips would be very welcome. 


I'm trying to solve some problems in dynamical systems that rely on this kind of estimate. Apologies if this is trivial.



Expectation and Dependence
This seems to be false.


Let $n=6$, $\ p=1/2$,
$$
A\_0 = 
\left(
\begin{matrix}
0 & 1 \\
0 & 1 \\
\end{matrix}
\right),
\ \
A\_1 = 
\left(
\begin{matrix}
1 & 0 \\
0 & 0 \\
\end{matrix}
\right).
$$


So the norm of the product is either $\sqrt{2}$, 1 or 0. (If the norms or matrices seem artificial, they can be adjusted by $\epsilon$ without affecting the result.)


The two expectations are $(720+127\sqrt{2})/4096 = .2196$ for the uncorrelated case and
$(724+126\sqrt{2})/4096 = .2203$ for the correlated case. 


