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Topic: Brownian motion sample path: fake or genuine?
Replies: 7   Last Post: Apr 3, 2013 8:12 PM

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 David Bernier Posts: 3,892 Registered: 12/13/04
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 : 2k-1205 in
>> the range {-1205, -1203, ... -1, 1, 3, 5, ... 1203, 1205}.
>> This number, 2k-1205, 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.

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 time-step (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 short-time variations of large amplitude
be under-abundant? 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

Date Subject Author
4/3/13 David Bernier
4/3/13 David C. Ullrich
4/3/13 David Bernier
4/3/13 Dan Heyman
4/3/13 David Bernier
4/3/13 David Bernier
4/3/13 David Bernier
4/3/13 David Bernier