I am trying to find the asymptotic behavior of the sum:

$$ \sum^n\_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$

as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|\leq1$ and $|y|\leq1$. I also know that $|x+y|\leq1$ and $|4xy|\leq1$.

One of the approaches I thought about was to approximate the sum by an integral. However, as it turns out, this approach runs into trouble since the terms in the sum do not give a sufficiently smooth function (I think: we have complex numbers raised to non-integer powers thus giving phase ambiguities).

On the other hand this sum may be related to Hoeffding's inequality applied to binomial distribution. This inequality gives an exponential upper bound. However, since $x, y$ are complex and $|x+y|\leq1$, I don't know how to apply this inequality.

Is there a way to bound this sum? I need to show it decays to $0$ as $n\rightarrow\infty$.