I am looking for an asymptotic upper bound or approximation on the probability that two i.i.d. multinomial ($n$,$\mathbf{p}$) with $Q$ categories are the same. Formally, I am looking for a way to bound the following: $$\sum_{x_1+\ldots+x_Q=n}\binom{n}{x_1,\ldots,x_Q}^2 \left(\prod\limits_{i=1}^n p_i^{x_i}\right)^2$$
Or, the simple Binomial(n,p) version of the problem
$$\sum\limits_{i=0}^n\binom{n}{i}^2 p^{2i} (1-p)^{2(n-i)}$$
When the distribution is uniform, from Theorem 4 in this paper (thanks to md5), the former becomes $$\sum_{x_1+\ldots+x_Q=n}\binom{n}{x_1,\ldots,x_Q}^2 Q^{-2n} \sim Q^{Q/2}(4\pi n)^{(1-Q)/2}$$ and making simpler arguments, the latter becomes \begin{align*} \sum\limits_{i=0}^n\binom{n}{i}^2 2^{-2n}&=\binom{2n}{n}2^{-2n}\\ &\le \frac{2\sqrt{\pi}}{e^2} \frac{1}{\sqrt{n}} \end{align*} both diminishing with $n$.
I have tried the approach given in the paper, but due to the asymmetry in the probabilities I could not obtain a fairly nice term, given in Eqn 5 in the linked paper. I tried entropy bounds as well, yet they gave me the trivial upper bound of 1.
I'd appreciate any help.