Limit of double sum involving binomials I am trying to get a meaningful interpretation of the behaviour of the following double sum in the limit of a large $t$ and small $p$. I look at its value as a function of $\beta$.  Here's the sum:
$$\sum_{j=0}^{\beta t} \sum_{k=0}^{j-1} {\beta t \choose j} {(1-\beta)t \choose k} p^{t-j-k} (1-p)^{j+k}$$
where $\beta$ takes only values such that $\beta t \in \mathbb{N}$.
It seems to me that the limit for fixed finite $p$ is a step-function (in $\beta$) with step at 0.5. 
If I compute numerically for large $t$ and small $p$ I get something that looks like the cumulative distribution function of a normal or perhaps laplace distribution (as a function of $\beta$).
Is there some meaningful way to read this as an integral in the appropriate limit? Or can you give me an exact description of 
$$
f(\beta) = \lim_{t\rightarrow \infty, p\rightarrow 0} \sum_{j=0}^{\beta t} \sum_{k=0}^{j-1} {\beta t \choose j} {(1-\beta)t \choose k} p^{t-j-k} (1-p)^{j+k}$$
Thanks!
 A: Consider some independent random variables $S$ binomial $(\beta t,p)$ and $R$ binomial $((1-\beta) t,p)$, then the double sum is $$P(R\lt S).$$
To show this, expand the event $[R\lt S]$ according to the values of $R=k$ and $S=j$, and use independence. 
Asymptotics follow easily, but the regime $t\to\infty$, $p\to0$ should be made more precise. For example, for every fixed $p$, $$\lim_{t\to\infty}P(R\lt S)=\mathbf 1_{\beta\gt1/2}+\frac12\mathbf 1_{\beta=1/2},
$$
by the law of large numbers if $\beta\ne\frac12$ and by the central limit theorem if $\beta=\frac12$ (and this might be the result you have in mind), but, for every fixed $t$, when $p\to0$, $R$ and $S$ both converge in distribution to $0$ hence $$\lim_{p\to0}P(R\lt S)=0.
$$
A: Since I was not only interested in asymptotic behavior, but also what happens when $t$ is 'large', another helpful way of looking at this, is to consider that for large $t$ and $p$ neither close to 0 nor 1, $B(t,p)$ can be approximated by a gaussian $N(tp, p(1-p)t)$.
The probability $P(R<S)$ can then be written down quite compactly using the error-function.
The poisson approximation is appropriate when $tp$ is fixed and $t$ large (as Did pointed out). In this case $P(R<S)$ can be calculated using the Skellam distribution.
