Consider the parameters $v_{a}$, $v_{b}$ be such that $0<v_{a}\leq v_{b}$ and $c>0$. I have an equation involving the Binomial distribution that I need to solve with respect to $p(n)$:
$\sum_{k=1}^{n}\left(\begin{array}{c} n-1\\ k-1 \end{array}\right)p^{k-1}\left(1-p\right)^{n-k}\cdot\left\{ v_{a}\left(\frac{1}{k}-\frac{k-1}{c\,k^{2}}\right)\right\} =\sum_{k=1}^{n}\left(\begin{array}{c} n-1\\ k-1 \end{array}\right)\left(1-p\right)^{k-1}p^{n-k}\cdot\left\{ v_{b}\left(\frac{1}{k}-\frac{k-1}{c\,k^{2}}\right)\right\} $
I can show that this solution has a unique solution $p\left(n\right)\in\left(0,1\right)$. (Assume that $c$ is such that each terms in brackets are positive for all $k$.) I wish to understand the solution to this equation, $p\left(n\right)$, in the limit $n\rightarrow\infty$.
I conjecture that $\lim_{n\rightarrow\infty}p\left(n\right)=\frac{1}{2}$. Any way to show this? Hint: The terms in brackets strictly decrease in $k$ and as $n$ grows the Binomial coefficients shift to the ``right'', so each of these sums should go to zero. Is this enough? I am concerned with the rate they go to zero... Thank you in advance for any inputs.
The answer to the question is that $\lim_{n\rightarrow\infty}p\left(n\right)=\frac{v}{v+1}$, where $v=\frac{v_{a}}{v_{b}}$. That is, my conjecture is correct, if and only if, $v_{a}=v_{b}$. (Existence and uniqueness) $p\left(\cdot\right)$ is bounded and strictly increasing. Hence, it converges to its supremum, which is unique. To explicitly find the limit $\lim_{n\rightarrow\infty}p\left(n\right)$, one has to use the fact that the LHS is the expectation of indenpendent non-IID $\text{Bin}\left(n,p\left(n\right)\right)$ and check that a suitable version of the CLT applies.