Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in distribution and determine the limiting distribution. Also need to determine if $X_n$ converges in other senses.

I know in general, the Bernoulli Characteristic function is \begin{align} \varphi(t) = 1 - p + pe^{it} \end{align}

But beyond that, I am really lost on how to use it to show convergence.

• Hints: 1) if $\psi(t)$ is the CF of some $X$, which is the CF of $a X$? 2) which is the CF of $X_1+X_2$, if $X_1,X_2$ are iid? – leonbloy Sep 26 '13 at 2:25

Bernoulli RVs are bounded, so you have that the convergence even occurs pointwise, hence a.s. pointwise, hence in probability, hence in distribution. Judging by the way the question was asked, I think your instructor wants a very specific answer. The limiting RV in any of the senses above is uniformly distributed on $[0, 1]$. You can sort of tell that's the case because if you think about what you're doing, at each step you are specifying the nth binary expansion digit, and you are doing so with equal probability for 0 or 1. To prove this rigorously, I think but am not completely sure that you can imitate the proof of the convergence condition for an infinite product. To state that heuristically, you basically want to compute the log of the product you are taking, do a Taylor expansion, show that second order and higher contributions disappear in the limit, and then you're done.