Lévy's forgery theorem

I asked my students in the exam of my graduate probability class to prove Lévy's forgery theorem which, in folk terms, states that if you run a Brownian motion long enough then it will write your signature (provided your signature is performed continuously). The one-dimensional version of this can be stated in mathematical terms as follows.

Let B be a Brownian motion and let f be a continuous function (your "signature"). Assume B(0) = f(0) = 0. Then, no matter how small ε > 0 is we will for sure be able to find a time interval of length 1 such that B will differ from f on that time interval by at most ε.

To prove this, all we have to prove is that there is a positive probability that the above will happen on the time interval 0 ≤ t ≤ 1, and then appeal to ergodicity (a theorem in probability): if you toss a coin sufficiently many times, you will for sure bring a head, provided that the probability of a single head is positive (assuming that coin tosses are independent).

The idea for proving the above is as follows. Split the interval 0 ≤ t ≤ 1 into n subintervals I₁, ... , I_n of length 1/n each and pick n so large that the maximum change of f within any interval of length 1/n is at most ε. (This can be done because f is uniformly continuous.)

Then observe that on each interval  I_k, the maximum deviation of B from its value at the beginning of the interval is smaller than ε with positive probability. This requires understanding that the maximum of a Brownian motion has density which leaves no gaps.

Also, observe that the difference of B and f at the beginning of the interval I_k can also be made smaller than ε with positive probability because when we observe a Brownian motion on equally spaced times, such as 0, 1/n, 2/n,..., then we are observing a random walk with normal increments.

Putting these things together we obtain a proof.

Intuitively, the proof was based on the following observation: The maximum difference of B(t) from f(t) will be small if 
(i) the maximum deviation of B(t) from its value at the beginning of the interval I_k containing t is small,
(ii) the difference between B and f at the beginning of this interval is small, and
(iii) f does not change much on this interval.

You might say, well, yes, this result is correct, but I might have to wait really long time to see my signature. And you'd be right. The analogy is this: if the probability of heads is 10^-10 then you will have to wait on the average  10¹⁰ billion seconds (more than 300 years) to see a single head, if you toss one coin per second.

We spoke of  "an interval of length 1'' above. Why? Is there any reason? No, absolutely none. By scaling, we can prove the same thing for any length. But this means that you don't have to wait too long, provided you don't mind if your signature is written in tiny letters. That is, the following holds:

When you wake up in the morning, pick a particle and make it move like a Brownian motion. Arrange things so that the particle's motion is monitored by a computer which keeps track of all places it visits. Go make a cup of coffee and come back and look at the computer files. The trajectory of the particle is stored in there. Zoom in, and zoom, and zoom, and zoom, and move around, and you will see your name. In fact, you will see the whole Bible, both in Greek and in Hebrew.

Is that amazing or what? You may say, "I don't believe this". This is a false statement. It's not a matter of belief. It's a matter of proof. Assuming that the hypotheses hold, the conclusion is true. The catch is, of course, that the hypotheses are mathematical ones and whether they are met in reality is a different matter. Of course, there is no such thing as "complete independence'' in real life, and there is no such thing as ``particle with infinitesimal size'' in real life. (Also, there is no such thing as a straight line...) However, a physicist can assure us that, with high precision, the assumptions are not unrealistic, meaning that, yes, even in real life the above claims are true. When it comes to seeing your signature some time far in the future, yes, you will see it, but it will take long time. When it comes to seeing your signature before you have finished your cup of coffee, all you'll see is black space because when you zoom and zoom and zoom, after a while you'll hit the limit imposed by the particle's size.

There is no problem with mathematics. There is no problem with physics. However, there is a problem with the word "belief". Simply, the word does not exist.

The gender of god

It's been long time since I posted anything, but time is scarce. Today, however, I was motivated by the silly response of someone identifying herself as Shirley J. Schultz in this posting from Sabio Lantz's Triangulations. She kept talking about god as if god were male:

... you also keep calling God “your God”. There is only one God. Different cultures call Him by different names. He is your God, too...

I don't understand why religious people (muslims, christians, jews, and others) don't ever wonder why they refer to god using the masculine personal personal pronouns "he", "him" and "his". Is it because they think god is male?

I raised this issue with muslims several times. Allah is gender-less, they respond. Fine then, I'll call Allah a "she". No, they reply. This is blasphemy.

Why can’t  religious people see the irrationality of attaching a male pronoun (and, therefore, a gender) to god? Let us consider the following two statements:

1. It is a fact that god has (male) gender because the scriptures tell us so and we may not refute the scriptures.
2. God has male gender because the concept was created by people in patriarchal societies.

Then statement (2) is much more likely than (1). If we were, therefore, to choose between (1) and (2) we ought to choose (2). Of course, both might be wrong, in which case a third option is needed. Since religious folk, like Shirley J. Schultz, seem to be firmly convinced that god has a (male) gender, they should either tell us whether they  accepts (1) or (2) or a third alternative which they should provide us with.

My fault, however, is that I tend to think that all people are rational. This is wrong. I devised the follloing hypothetical dialogue with  Shirley J. Schultz:

Shirley: God does not have gender. He is above human nature.
Takis: But you keep using “he”.
Shirley: Well, this is just a convention.
Takis: OK, I understand. So it’s fine if I say “God’s works reveal her glory”.
Shirley: No, you can’t say that.
Takis: Why not?
Shirley: It’s a blasphemy.
Takis: But you told me god does not have gender.
Shirley: Yes, but the bible says “he”.
Takis: Was the bible written by god?
Shirley: Yes, of course!
Takis: Then god tells us god prefers to be addressed by the male pronoun.
Shirley: This is not important. He does not have gender.
Takis: OK, may I then refer to god as “it”? Like: “God’s works reveal its glory”.
Shirley: You may not (emphatically) call god “it”. God is not a thing!
Takis: I didn’t say so, I was just trying to avoid assigning a gender to god. So, if “it” is not allowed, I will use “she”.
Shirley: No, you can’t do that.
Takis: Why not?
Shirley: It’s a blasphemy.
Takis: But you told me god does not have gender.
etc.
etc.

There is absolutely no way to get the argument resolved, because Shirley has no way to defend her position. She is falling into a contradiction. On one hand god has no gender, but on the other we are being told god likes to be called a “he”. So, even if god has no gender, god prefers the male pronoun.

A second version of the dialogue (I won’t bother writing it down) could start by Shirley supporting the idea that man is created in the image of god, so god has a gender. Now, why god has male gender is baffling, and Shirley won’t be able to defend the simple observation that this implies that god gives preference to men.

In either case, Shirley has no rational argument and/or will reach a contradiction. But such is the religious mind. It is not disturbed by the irrational even when the irrational is obvious. The religious mind would rather hurt itself or hurt other but would never admit violation of (elementary) logic.

Sabio gave me a response. He thinks that Shirley would have responded in this way:

“Jehovah, the one true God, has always been spoken of a as a masculine entity. God is certainly above our understanding, and this use of a masculine pronoun should not narrow our thinking, but it is the closest pronoun to describing God than any other. To assume God is anything like your common notions of the word “he” just illustrates how dimly you see into the nature of God. He is there for you, Takis, be open to his spirit and language will stop becoming an obstacle between you and God.”

 And continued, thus:

Shirley could have mustered together a fairly intelligent response, or she could have fumbled and given a totally contradictory rambling. But if you think you are going to trap her and it will change her, you may be wrong if she is more highly vested in her commitment than to a temporary argument. Even if you frustrate her argument, she will come back with some trite platitude and tell you how you must come to worship the one true God.

He is right. I still, however, would love a rational response  about god's gender from Shirley J. Schultz or other religious people.

P.S. Soon I hope to post another article, related to gender issues which are very important in Sweden.  As I pointed out before, they even have a government department dealing with these issues. A few months ago, I attended a couple of lectures on gender issues, and, in particular, on issues regarding women's under-represented positions, and asked the question whether they have ever considered the role of religion in connection with this. Surprisingly, the speaker didn't have a clue and hadn't even thought about it, apparently. More on this in a future post.

Another prestigious conference....

A colleague recently forwarded to me the following call for papers. It is, undoubtedly, one of the funniest announcements I've ever seen. What I like most is that they welcome submissions in all academic fields! Also, notice that the deadline for submission of papers is 3 weeks before the start of the conference!

EMF 2013, EURASIAN MULTIDISCIPLINARY FORUM,
Tbilisi, Georgia 24-26 October 2013

SUBMISSION DEADLINE: 1st October 2013

Distinguished Colleagues,
    We have the pleasure of inviting you to submit your scientific work for the EMF, Eurasian Multidisciplinary FORUM, 24-26 October 2013 which will be held at the "Grigol Robakidze" University, Tbilisi, Georgia.
    All accepted papers will be published as a special publication with an ISBN number. The author will be sent a printed copy of the publication after the conference finishes.
    The papers will be also published in a special edition of one of the most Influential international scientific journals in South Europe, The European Scientific Journal (ESJ).
    Authors of the accepted papers that will not be personally present on the event will be given the opportunity, on their request, to make an online presentation for the conference participants.
    The authors are not obliged to present their works (personally or online).
    EMF 2012 is organized by: European Scientific Institute, "Grigol Robakidze" University and the  University of The Azores, Portugal.
    Supporting the concept of interdisciplinarity we welcome submissions in all academic fields.
    Sincerely,
    European Scientific Institute
    "Grigol Robakidze" University
    University of The Azores

Color Photographs of Greece, 1920’s

This is taken from here. Click for more photographs.

YouTube: searching for playlists

As is well-known, YouTube is a good medium but it is surprisingly monolithic, difficult to use, hard to organize. This is, of course, not a coincidence. The hackers of YouTube are no idiots. Most likely, they have clear instructions to make the system inflexible so that people (you and I) keep always clicking for new videos. Their dominant term in their utility function is the maximization of clicks by the average user, in the hope that the user will become victim of some advertisement once in a while.

The inflexibility of YouTube includes:
-- inability to organize playlists in different order (say reverse chronological order)
-- impossibility to shift items from one playlist to another
-- very limited search
-- impossibility to search for playlists

However, there are some things you can get around, and here is my one-time contribution to the art of hacking: Here is how to search for playlists.

Copy and paste the following into the URL field of your browser

https://gdata.youtube.com/feeds/api/playlists/snippets?q=KEYWORD&start-index=N&max-results=M&v=2

after replacing the items in yellow by what  you need them to be:

KEYWORD : a search term, case-insensitive
N : start index, an arbitrary positive integer
M: number of items to be listed, an integer, 1 ≤ M ≤ 50

 N.B. No, this is not my thing. I got it from here.