Ok, this is the LAST email I will submit on this subject. Either I
will convince you of the validity of my ideas, or I will give up. I
do appreciate everyone giving me the chance to prove my point. All of
this reminds me of preparing for my Phd Thesis defense some years
ago. It took awhile back then, but ultimately I did prevail.
···
----
Just a couple of things though: I really emplore everyone who had
something to say about the CLT to read this entire post. If you have
any questions, please go ahead and ask me (you can use private email)
or contact the authors of the works I reference directly. I'm sure,
as acedemics, they would be happy to discuss this topic with you.
-----
Since I got a lot push back over my quick and dirty prior attempts, I
figured this time I would do a more rigorous job. That said, I will
avoid formal proofs, which can be found in the references given. I am
also going to use as few references as possible. Here they are:
1) A series of lecture notes for an MIT math course 18.366 Random
Walks and Diffusion. The notes are available on the web at
http://www-math.mit.edu/18.366/lec/ . These notes have titles
like: "3: The Central Limit Theorem.", "4: Asymptotics Inside the
Central Region" and "5: Asymptotics with Fat Tails." I urge
everyone who has chimed in on this debate to take a look at the
lecture notes, especially Lecture 4, which discusses the non CLT
behavior of distributions with fat tails in detail. I call this
reference "the lecture"
2) A recent 2004 book entitled "Elements of the Random Walk" by
Joseph Rudnick and George Gaspari. I don't think the text of the
book is available on the web, so I scanned in the relevant pages and
can send them to anyone who is interested. It is a good book (a
review can be found here http://edge-
online.org/pdf/tle2406r06640665.pdf ), This book is also worth a
look, especially section 8.1.1 titled "Non-Gaussian behavior" You
can also buy the book
(http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521828910
or at Amazon) or get it at your library (MIT has a copy). I call
this reference "the book"
Here is a list of what I am going to discuss. [ I am going to avoid
some of the concepts I brought up before (such as symmetry, etc), but
those who are interested can read about them in the book or ask me
later.]
1) Random Walk Processes
a. Simple example: flipping a coin (an unbiased process)
b. The variance of all simple (unrestricted) processes is linearly
proportional to the number of steps (or coin flips)
c. Most (but not all) Random walk processes have Gaussian or normal
PDFs in the limit of large number of steps
2) VP as a random Walk Process (with some bias)
a.VP is an unrestrained process, so variance is linearly proportional
to number of hands
b.VP probability table falls off quite slowly
its not like a coin
flip
it has a fat tail
c.CLT does not hold for such distributions, PDF is not normal, though
central section might be (without a stationary central limit though,
due to bias)
So here goes: BTW, I use the abbreviation RA for reference available
(just ask me).
So called Random Walk (RW) processes have been studied for a long
time (lecture 1). Though many believe that RW processes were first
studied because of their connection to gambling (RA), however this is
not likely the case. A result of a series of coin flips is an
example of a simple RW. For example, an idealized 2 sided coin is
tossed a number of times, say 5. Each time the coin comes up heads,
you take a step the Right and each time the coin comes up Tails you
take a step to the Left. After the 5 coin tosses, you would have
taken 5 steps. But the position you end up at could be anywhere in
between 5 steps to the Right and 5 steps to the Left of where you
started, even though the average result is that you end up just where
you started. Instead of tossing the coin 5 times you could toss it
100 or 1000 times. In each case, you end up between S and +S steps
away from where you started, where S is the number of coin tosses.
Once again, the average result is that you end up where you started.
The variance of such a process is linearly proportional to the number
of coin tosses, and because of the Central Limit Theorem, the
Probability Distribution Function for this RW process obeys "Gaussian
statistics"that is it becomes normal as the number of coin tosses
gets large (RA). Because this was an "ideal" two sided coin, it has
no "bias". Hence, independent of the number of coin tosses, on
average you end up just where you started (this isn't really true for
a small odd number of tosses, but it doesn't matter here). That is,
there is no systematic, long term drift in the PDF (the mean doesn't
change, its' always at 0, ignoring the small-odd exception). It is
well known that there are many RW processes that behave like this
that is, show convergence to a normal distribution, as the central
limit theorem (CLT) would predict. However, it is also well known
that many RW processes do not converge to a normal distribution (see
book and lecture 1 page and lecture 4 for examples).
VP play can be treated like a RW. Unlike the ideal two-side coin,
VP games have multiple possible outcomes or "steps" for each "draw"
(or toss). For example, the VP game "Jacks or better" has 10
possible outcomes for each draw. These include taking a zero step
(0) to taking a 4000 long step (for a Royal Flush). If one includes
the amount bet, these steps can range for JoB between a step of -5
to a step of 3995 (assuming 5 units are bet). Even though the JoB
VP game has 10 possible outcomes, we can treat multiple draws of it
just like we did for the coin toss, but instead of actual walking
left or right, we will just total up the outcomes. After 5 draws
for example, your total outcome will be somewhere between -25 to
19975. Surely not all outcomes have equally probability. For
example, it is far, far more likely to end up at -25 than at 19975.
Interestingly, the average outcome of 1 draw of JoB is about -0.023
[or (99.954-100)% of 5]. This means that JoB has a net negative
bias. Hence as we play more and more hands, we will tend to walk
towards one side, to the losing side unfortunately. In other words,
though we are on a random walk, we have a tendency, on average, to
walk one way more than another.
Since JoB (and all random VP games with independent draws) is a
simple RW process, its variance is proportional to the number of
draws or hands played. That means, like all other RW processes, that
the variance increases without limit as the number of hands
increases. This has been proven in the literature many times (see
book, lecture notes, and variance data at `The wizard of odds' VP
website). That said, what does the PDF of the process look like?
Could it, like the coin toss example, have a normal PDF due to the
CLT? Well, for starters, whatever it looks like, its average value or
mean drifts slowly to the left, at a rate of -0.023/hand.
Now this is where it gets complicated. In order for the CLT to hold
for RW processes, the probability of an arbitrarily long step must be
very low. That is, as the (magnitude of the) step gets larger, the
probability of such a step occurring must get small very quickly.
Mathematically, it is known that the probability needs to get smaller
faster than (magnitude of the step size)^(-5) if the CLT is to hold
(book page 194, equation 8.1) . This is no problem for the coin toss
RW. There are only two possible steps, -1 or +1 unit, and each has
the same probability. All other steps occur with exactly zero
probability. Hence the probability of an arbitrary large step (or any
step greater than 1) falls off infinitely quickly. This is why the
CLT holds for the coin flip. But the situation for VP is very
different. VP is characterized by many small or smallish steps and
only occasionally (or rarely) a very large step. It is well know
that such RW generally exhibit non CLT behavior. For an example see
Lecture 1, page 5, which I quote from below:
"One way to violate the CLT with IID displacements is via 'fat-
tailed' probability distributions, which assign sufficient
probability to very large steps that the variance is infinite
..We
see that [in this case] the walk mostly takes small steps, but
occasionally makes very large jumps, comparable to the total
displacement."
Mathematically, it can be shown that for JoB, the probability falls
off at approximately (step size)^(-1.7). This is no where near the
limit for CLT to hold (book, page 194, eq 8.1). Other VP games
behave in a similar manner, and hence their PDF's are not normal,
regardless of the number of hands played.
At this point, I believe I have firmly established that the PDF of
JoB (or VP in general) is not ever normal because the CLT does not
apply. Though this may seem an unbelievable result to many of you,
it actually is a quite common occurrence in natural (physical)
systems. Hence, such non-CLT behavior has been thoroughly studied in
a range of diverse fields including physics, meteorology, and
economics.
That said, the PDF for such non-CLT processes develops what is called
a "central region", which has Gaussian-like behavior (see Lecture
note 4). This is indeed what happens with VP. However, since VP is
a biased RW process, the behavior of the central region is
complicated, since it drifts towards smaller and smaller values as
the number of hands increases.
Ok... so if you don't beleive me now (or the authors of the book or
lecture notes), I guess you never will.