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Let $a,b,c$ be three positive integers. Suppose that $X_a,X_b,X_c$ are three mutually independent binomial variables satisfying that $X_\lambda\sim B(\lambda,\frac12),\lambda\in\{a,b,c\}$. Here $X\sim B(\lambda,\frac12)$ means $\mathbb P(X=k)=2^{-\lambda}\binom{\lambda}{k}$ for $k=0,\cdots,\lambda$.

Let $S=X_a+X_c$ and $T=X_b+X_c$.

I'm interested in estimation (upper bound, especially) of their joint distribution $$F(s,t):=\mathbb P(S\le s,T\le t)$$ when $\min(a+c,b+c)$ is large. Formally, I want to obtain an upper bound (accurate to $O({[(a+b)(b+c)]^{-1/2}})$ if one can) of $$2^{-c}\sum_{k=0}^{c}\binom{c}{k}\mathbb P(X_a\le s-k)\mathbb P(X_b\le t-k).$$

Perhaps this problem has been solved by other researchers. If so, it would be very nice if someone could give me some references.

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