
Re: Brownian motion sample path: fake or genuine?
Posted:
Apr 3, 2013 10:36 AM


On 04/03/2013 09:40 AM, dullrich@sprynet.com wrote: > On Wed, 03 Apr 2013 03:17:45 0400, David Bernier > <david250@videotron.ca> wrote: > >> I used snow (white noise) from an analog cathode ray tube TV, >> recorded with a webcam, as the source of entropy for supposedly >> random bits. >> >> I had 1.8 gigabytes of digital video, which I >> whittled down to about 2.5 megabytes of "random bits". >> >> 16384 Bernoulli(1205, 1/2) r.v.s were generated, >> with k in {0, ... 1205} associated >> to the #0 s minus #1s : 2k1205 in >> the range {1205, 1203, ... 1, 1, 3, 5, ... 1203, 1205}. >> This number, 2k1205, stood in for a Gaussian r.v. of >> mean zero. The result was a random walk with steps >> of +1 and 1, sampled every 1205 steps; >> 16384 samples were taken, without counting >> x(0) = 0 [starting point]. >> >> >> 16384*1205 = 19,742,720 (bits) or 2,467,840 bytes. >> >> The graphic drawn with MatLab is here: >> >> >> http://img203.imageshack.us/img203/9220/brown1.png >> >> >> Maybe it's a fake, not random. >> I don't know ... > > You don't say so explicitly, but it sounds as though > you feel that the picture looks wrong somehow. > It looks like Brownian motion to me  what seems > to be the problem? (Maybe I'm missing your point...) [...] In a first stage, I wasn't concerned and was really alluding (in jest) to a conspiracy, or "the revenge of the electrons" in the TV tube ...
Alternatively, ghosts, demons, aliens, spies or the "Improbability Drive" in the Hitchiker's Guide to the Galaxy could be corrupting influences on the bits from webcam recordings of the CRT TV tube.
In midphase, I had doubts about my C bitextraction program (bugs): this is a minor concern.
In the final stage, I considered that if `K' follows a Bernoulli(1205, 1/2) distribution, then 2K  1205 is a discrete probability distribution with support: {1205, 1203, 1201, ... 1, 1, ... +1205}, hence not a continuous Gaussian r.v. of mean zero, as in the real Wiener process.
I would have to do further computations to satisfy myself whether or not artifacts of discretization using 1205 "coin tosses" per timestep (with 16,384 time steps) are likely to remain to a very determined examiner.

I had a look at the Brownian motion (formally Wiener process) sample path at Wikipedia:
http://en.wikipedia.org/wiki/File:Wiener_process_zoom.png
Is it just me, or might shorttime variations of large amplitude be underabundant? To me, it seems like overall, "dramatic" drops and rises (very steep) are less pronounced at Wikipedia:
http://en.wikipedia.org/wiki/File:Wiener_process_zoom.png
than at:
http://imageshack.us/f/203/brown1.png/
There, there's a very steep rise for t around 6700 to 6800, with the position x(t) crossing the horizontal axis, x = 0.
David Bernier

