Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [-1,1]$ for $n=0,1,\dots$ is a sequence of functions that are $L$-Lipschitz with regards to the max metric on $[-1,1]^n$, i.e. $$|f_n(x_0,\dots,x_{n-1})-f_n(y_0,\dots,y_{n-1})|\leq L \max\{|x_0-y_0|,\dots,|x_{n-1}-y_{n-1}|\}$$ for any $n$ and $x_0,\dots,x_{n-1},y_0,\dots,y_{n-1}\in [-1,1]$. Note that each of these functions interpreted as a function on the $\ell^\infty$ unit ball is weak* continuous and therefore the function $f(x_0,x_1,\dots)=\sup_nf_n(x_0,\dots,x_{n-1})$ is weak* semicontinuous (and in particular measurable). Also note that $f$ is still $L$-Lipschitz with regards to the $\ell^\infty$ norm.
Let $Y=f(X_0,X_1,\dots)$ be the random variable given by applying $f$ to the sequence $X_i$. I want to be able to say that $Y$ cannot be too sharply bimodal with a bound given by $L$. To be more precise I want to say that there is some $r(L)<\frac{1}{2}$ such that $\min\{P(Y\leq\frac{1}{3}),P(Y\geq \frac{2}{3})\}\leq r(L)$ for any sequence of $L$-Lipschitz functions $f_n$.
Does such an estimate exist for every $L$?
