For $v, w \in \{0,1\}^n$, denote $v w = (v_1 w_1, \ldots, v_n w_n)$ and $|v|=\sum_{i} v_i$.

Let $v_1, v_2 \in \{0,1\}^n$ and \begin{align*} f(x_1, x_2) = \sum_{d=0}^{|v_1 v_2|} \frac{1}{2^{|v_1|+|v_2|-|v_1 v_2|}} {|v_1| - |v_1 v_2| \choose x_1 - d} {|v_2| - |v_1 v_2| \choose x_2 - d}. \end{align*}

How to find the asymptotic formulas for $f(x_1, x_2)$ ($|v_1|, |v_2| \to \infty$)?

Is it possible to prove that there is a constant $\epsilon$, $0<\epsilon<1$, such that \begin{align} \sum_{x_1=0}^{|v_1|} \sum_{x_2=0}^{|v_2|} f(x_1, x_2) < 1 - \epsilon? \end{align}

These questions relate to the question.

Any help would be greatly appreciated!