The following is a qualifying exam problem.
Let $\{X_n\}$ be an i.i.d. sequence of Binomial($n,1/2$) random variables. Define the sequence,
$$ Y_n = \left(1+\frac{1}{\sqrt{n}}\right)^{X_n}\left(1-\frac{1}{\sqrt{n}}\right)^{n-X_n} $$
Find the limiting distribution of $Z_n = \ln(Y_n)$.
I tried using a characteristic function approach. The chf of $Z_n$ is given by,
$$ \begin{eqnarray*}\phi_{Z_n}(t) &=& E[e^{itZ_n}] \\ &=& E[e^{it\ln(Y_n)}] \\ &=& E[e^{itX_n\ln\left(1+\frac{1}{\sqrt{n}}\right)}e^{it(n-X_n)\ln\left(1-\frac{1}{\sqrt{n}}\right)}]\\ &=& \sum\limits_{k=0}^ne^{it\ln\left(1+\frac{1}{\sqrt{n}}\right)k}e^{it\ln\left(1-\frac{1}{\sqrt{n}}\right)(n-k)}\binom{n}{k}\frac{1}{2^n}\\ &=& \left(\frac{(1+\frac{1}{\sqrt{n}})^{it}+(1-\frac{1}{\sqrt{n}})^{it}}{2}\right)^n \end{eqnarray*} $$
A computation in Mathematica reveals that,
$$ \lim\limits_{n\rightarrow\infty}\phi_{Z_n}(t) = e^{-\frac{1}{2}it - \frac{1}{2}t^2} $$
which we recognize as the chf of a Gaussian distribution with mean $-\frac{1}{2}$ and variance $1$.
A couple questions:
1) Is there a straightforward way of computing the limit above (without the use of CAS software)?
2) Is there a more time efficient approach here? Keep in mind that this was a qualifying exam problem.
You can use the central limit theorem: \begin{align} &\ln Y_n \\ & = X_n \ln \frac{1+\frac{1}{\sqrt{n}}}{1-\frac{1}{\sqrt{n}}} + n \ln \left(1-\frac{1}{\sqrt{n}}\right) \\ % % & = \underbrace{\sqrt{n} \ln \frac{1+\frac{1}{\sqrt{n}}}{1-\frac{1}{\sqrt{n}}}}_{\xrightarrow[n\to\infty]{} \, 2} \times \sqrt{n}\left( \frac{X_n}{n} - \frac{1}{2} \right) % + \frac{1}{2}n \ln \frac{1+\frac{1}{\sqrt{n}}}{1-\frac{1}{\sqrt{n}}} % + n \ln \left(1-\frac{1}{\sqrt{n}}\right) \\ % % % & = \sqrt{\mbox{Var}(X_1)} \underbrace{\sqrt{n} \ln \frac{1+\frac{1}{\sqrt{n}}}{1-\frac{1}{\sqrt{n}}}}_{\xrightarrow[n\to\infty]{} \, 2} \times \frac{\sqrt{n}}{\sqrt{\mbox{Var}(X_1)}}\left( \frac{X_n}{n} - \frac{1}{2} \right) + \frac{n}{2} \ln \left( 1+\frac{1}{\sqrt{n}}\right) + \frac{n}{2} \ln \left( 1-\frac{1}{\sqrt{n}}\right) \\ % % % &\xrightarrow[n\to\infty]{D} Z-\frac{1}{2} \end{align} where Var$(X_1)=1/4$, $Z\sim N(0,1)$, and in the last I used that \begin{align} \lim_{n\to\infty} \left( \left( 1+\frac{1}{\sqrt{n}}\right)\left( 1-\frac{1}{\sqrt{n}}\right)\right)^n = \lim_{n\to\infty} \left( 1-\frac{1}{n}\right)^n = \frac{1}{e} \end{align}