Randy Yates <randy.yates@sonyericsson.com> wrote in message news:<xxpzn1u9xzh.fsf@usrts005.corpusers.net>...
> allnor@tele.ntnu.no (Rune Allnor) writes:
>
> > Randy Yates <yates@ieee.org> wrote in message news:<7jp0bmy9.fsf@ieee.org>...
> > > allnor@tele.ntnu.no (Rune Allnor) writes:
> > > > [...]
> > > > The CLT as I know it, applies to PDFs, not random variables.
> > >
> > > Do you mean that the convolution it speaks of is of the PDFs, not
> > > the random variables themselves?
> >
> > That's exactly what I have been saying during this whole thread!
>
> No, that's not "exactly" what you've been saying, and that is part
> of my issue with you, Rune. You said, exactly,
>
> The CLT as I know it, applies to PDFs, not random variables.
>
> This statement does not mention convolution.
C'm on, Randy. If you read the whole thread (including your own
posts),
you will find that the OP convolved the data in the first place.
In fact, you yourself wrote
"Secondly, when you add independent random variables, the distribution
of the result is the convolution of the input variables' PDFs."
in the post of November 4th (your first post in this thread). We agree
in the basic properties of the CLT; why do you make such a fuzz about
disagreeing with me now?
> > > However, to
> > > say that the CLT does not apply to random variables is pretty
> > > much false in my book - it's all about random variables.
> >
> > OK, if we have to go nit-picking,
>
> If I'm nit-picking, then so is Papoulis. His section on the CLT
> begins like this:
>
> Given n independent *RVs* x_i, we form their sum
>
> x = x_1 + ... + x_n
>
> This is an *RV* with mean ... and variance ... . ... Furthermore, if
> the *RVs* x_i are of continuous type, ... the density f(x) of x
> approaches a normal density ... . This important theorem ... .
>
> [emphases mine]. He CLEARLY associates the CLT with RVs.
I have never contested that. But if you want the CLT to work and
produce Gaussian distributions, you need to work on the PDFs.
> Now it is true that he also goes on to say "The CLT can be expressed
> as a property of convolutions ...", but it seems pretty clear that the
> main interpretation and utility of the CLT is in association with RVs.
> To divorce it from RVs and speak only of convolving "positive
> functions," while theoretically accurate,
Make up your mind. Do you agree in tht what is convolved to produce
results according to the CLT are PDFs, or do you not agree?
> robs it of its real value:
> explaining why randomness in nature is often Gaussian.
No. The CLT is an ad hoc excuse for the analyst to stay with the
nice and easily tractable Gaussian distributions instead of diving
into the more tricky ones. The CLT does not "make a non-Gaussian
process
Gaussian", it only provides some comfort in stating that one does not
make a very big mistake if one chooses to work under the Gaussian
hypothesis.
> > here's my 2c: A "random process"
> > generates "random variables" (or "random data"), RVs, that in some
> > way are characterized by a "Probablility Density Function", PDF.
> > In that sense, the RV and the PDF are interconnected in that both
> > are associated with a random process.
>
> Wow. Now that's rich, Rune. After two courses in Random Processes
> and another two in basic probability theory, I've never heard anyone
> condition the association of a RV and its PDF on an association
> with a random process. I don't know where you've come up with that
> idea, but it is completely unorthodox in my experience.
Is a "random process" unorthodox to you? (OK, I should perhaps used
the
term "stochastic process", but I didn't want to go pedantic on you...)
Hey, Randy, this is a joke, right?
> > The "CLT operator"
>
> Huh? Since when was anyone talking about a "CLT operator"? You've just
> now introduced new language. The topic of discussion thus far has been
> about a theorem, the "Central Limit Theorem," NOT an operator!
I'm not introducing new language. If you take a course on linear
systems
in maths, you'll find the term "operator" used all over the place.
Particularly in the context of convolution integrals.
If you express the CLT as a property of the expression
y_CLT = y_1 (*) y_2 (*) ... (*) y_N
where (*) means convolution and y_n are PDFs, the term "CLT operator"
makes perfect sense.
> > takes multiple PDFs as input and produces one
> > PDF as output. When I look at the inner workings of the CLT, I see
> > PDFs, not RVs. I could have agreed with you if you said "it's all
> > about random _processes_". You didn't.
>
> No, I certainly did not, because the CLT (reverting to the terminology
> that we've been using) at least as presented by Papoulis, is not about
> a random process. It has NOTHING to do with random processes.
Well, you may disagree with my approach to these matters and the
exact way I interpret the problem and phrase my opionions. You should
be very careful about how you state your objections, though. You might
find yourself in a position you can not defend.
> > My point is that mentioning the CLT only makes sense when studying
> > PDFs.
>
> I heartily disagree, for the reasons I've already explained above.
Please, Randy, I know you don't mean this. Yes, the effects of adding
several random variables is the reason why the CLT is interesting.
Arguing *why* the CLT works, and *how*, requires the studying
stochastic processes and the convolution of their PDFs. Not the
random variables.
For the simple reason that given a random vector, you don't know
anything about its PDF. You can make up an opinion, based on a
histogram, but you don't know. The concept of a PDF only makes sense
in the context of a stochastic process.
> > The OP tried to link the CLT directly to the random variable.
>
> As well he should. The only problem is, he apparently did so
> improperly (i.e., via convolution of the RVs rather than the
> sum of the RVs).
The OP used random data (a single realization of a random variable)
where a PDF should have been used. The exact nature of the PDF was
never specified (not enven an estimate through a histogram), and
no histogram of the resulting data were used. The important difference
between a stochastic process generating random variables, and the
random data as a realization of sucha random variable, was never
grasped. The question was phrased in a way that disagreed just enough
with standard terminology to cause confusion (denoting the random
variable by the symbol "f", which usually is reserved for PDFs).
Apart from that, the OP did an excellent job in verifying the CLT.
Rune