TITLE: Exponentially weighted infinite sum of Bernoulli variables
QUESTION [9 upvotes]: Consider the following process.  For each integer $i \geq 0$, independently sample a Bernoulli distribution with probability $p = 1/2$, obtaining sample $x_i$.  Then calculate $x = \sum_{i=0}^\infty x_i \theta^i,$ where $(\theta < 1)$.  What is the distribution over $x$?
I see that if θ = 0.5 then this is a uniformly generated binary number between 0 and 2.  This post is closely related, but I don't see how the method given there (with θ = 0.5) generalizes to arbitrary $\theta$.  I am interested in values of $\theta$ close to 1, such as $\theta = 0.95$.
I am ultimately interested in a hypothesis test:  $H_0=$ "$p = 0.5$" against the alternative $H_A=$ "$p \neq 0.5$".  The motivation is, I am receiving an infinite sequence of 0s and 1s for which I only retain an exponentially weighted average (not recording the whole sequence).  And based on this weighted average I want to decide whether the 0s and 1s were generated with equal probability or not.
Edit: based on an empirical test it appears to be roughly normally distributed.  The following was generated by performing a 100,000 sample run with $\theta = 0.95$.  It has mean 10 and variance 2.5.
 5.9 
 6.0 
 6.1 *
 6.2 *
 6.3 *
 6.4 *
 6.5 **
 6.6 **
 6.7 **
 6.8 ***
 6.9 ***
 7.0 ****
 7.1 *****
 7.2 *****
 7.3 ******
 7.4 ******
 7.5 ********
 7.6 ********
 7.7 *********
 7.8 *********
 7.9 **********
 8.0 ************
 8.1 ************
 8.2 *************
 8.3 **************
 8.4 ***************
 8.5 ****************
 8.6 *****************
 8.7 ******************
 8.8 *******************
 8.9 *******************
 9.0 ********************
 9.1 *********************
 9.2 **********************
 9.3 ***********************
 9.4 **********************
 9.5 ***********************
 9.6 ***********************
 9.7 *************************
 9.8 ************************
 9.9 ************************
10.0 ************************
10.1 ************************
10.2 *************************
10.3 ************************
10.4 ***********************
10.5 ***********************
10.6 **********************
10.7 ***********************
10.8 **********************
10.9 *********************
11.0 *******************
11.1 *******************
11.2 ******************
11.3 ******************
11.4 ****************
11.5 ***************
11.6 ***************
11.7 *************
11.8 ************
11.9 ************
12.0 ***********
12.1 **********
12.2 *********
12.3 ********
12.4 ********
12.5 *******
12.6 ******
12.7 *****
12.8 *****
12.9 *****
13.0 ****
13.1 ***
13.2 ***
13.3 **
13.4 **
13.5 **
13.6 *
13.7 *
13.8 *
13.9 *
14.0 
14.1 

Normal quantile plot for $\theta=0.90$:

Normal quantile plot for $\theta=0.95$:

Looks like it has thin tails.

REPLY [4 votes]: This type of distribution has a long history. If you apply an affine map $x \mapsto 2x-1$, you may assume the Bernoulli variables take values $\pm 1$. Then the laws of these sums have been studied under the name "Infinite Bernoulli convolutions", see [1], [2] for surveys. The basic result is that for a.e. $\theta \in (1/2,1)$ the laws are absolutely continuous; the exceptions have Hausdorff dimension zero, and all known exceptions are algebraic numbers (in fact, inverse Pisot numbers.)
[1] Peres, Yuval, Wilhelm Schlag, and Boris Solomyak. "Sixty years of Bernoulli convolutions." In Fractal geometry and stochastics II, pp. 39-65. Birkhäuser, Basel, 2000.
[2] Varjú, Péter P. "Recent progress on Bernoulli convolutions." arXiv preprint arXiv:1608.04210 (2016).