Re: Basic Doubts regarding sequence of Random Variables and Stochastic Process?

On Mar 20, 7:21 am, Sujit <sujit.gu...@xxxxxxxxx> wrote:
Hello All,

I have couple of basic doubts regarding sequence of random variables/
stochastic process.

1a. What is the difference between sequence of random variables and
discrete time stochastic process defined
on same probability space?
1b. A = set of all sequence of Random Variables on some probability
space, say SPACE.
B = all stochastic process on SPACE.

Is A subset of B or B subset of A or neither or it doesn't make any
sense to compare A and B?

B is a subset of A. Usually one begins the definition of a stochastic
process by saying "A discrete-time stochastic process is a sequence of
random variables such that ...". Actually, this may be largely a
question of semantics, but of course, 'subset' does include the
possibility of set equality, so possibly there is no difference
(depending on the author, etc.).


2. Say I have sequence of random variables:
Probability space Omega = [0,1] (closed interval 0,1)
Uniform distribution.
X_1(w) = 1
X_2(w) = 1 if 0<=w<=0.5

What happens to X_2(w) for w > 1/2?

.
.
.
.
X_n(w) = 1 if 0<=w<=1/n

What is X_n(w) for w > 1/n?

.
.
.
.

Does X_n converge to 0 almost surely(a.s).

Is this a qu;estion? If so, use a question mark '?'. Anyway, if X_n(w)
= 0 for w > 1/n, then we have X_n(w) --> 0 for all w in a set of
probability 1, so the answer is YES.

If basic definition is
applied,
X_n converge to 0 a.s.
But Now if i apply Borel-Cantelli lemma, \sum P(X_n>\epsilon)
diverges,
so X_n should not converge in a.s.

Which Borel-Cantelli Lemma are you trying to apply? One of them
involves a finite probability sum as a /sufficient/ condition that
only finitely many events occur w.p.1; the other one involves an
infinite probability sum plus independence as a /sufficient/ condition
that infinitely many events occur w.p.1. Neither result is a
_necessary_ condition.

So only thing can go wrong is X_n are not independent, which is
also required to apply BC lemma.

No, the other thing that can go wrong is that you are trying to apply
a sufficient condition as though it were a necessary condition.


I am not getting, why X_n are not independent? Or X_n won't
converge to 0 a.s.?

3. I am not getting clear idea about sequence of random variables:
When some event occurs, we say X1,X2,...take some value (based on
definition of RV)
or its like at some different epoch, different RV come into
picture.

What on earth does this mean?

I got this confusion because, standard example is X_i is ith
burnolli trial's o/p.

I don't know what 'o/p' stands for. Please write thing out completely.

S_n = \sum{i=1}^{n} X_i /n.
So say, coin toss, and if head on ith coin toss, X_i = 1 else 0.

Now my doubt is, when we are talking about sequence of RVs,
Is it like, some event occurs and based on that event,
X_1,X_2,...,X_n,... takes values
OR
Some, event occurs, X_1 will take value depending upon the event,
again some event occurs,
X_2 takes some value based on event and so on...and this
collection will be representing
sequence of RV?

The general theory of (discrete) stochastic processes involves an
"infinite" sample space S, so that your sequence is X_1(s), X_2(s),
X_3(s), ... for s in S. It is as though s determines all the X_i at
once. Of course, in order for the X_i to be independent, or for X_i to
not depend on X_{i+1}, X_{i+2}, ... (the non-anticipating property)
you need some structure on the space S and some restrictions on the
functions X_i(s).


These are very basic doubts. But reading books, has not cleared me, so
if somebody can help me
understanding these concepts, I will be obliged,

Maybe you have been reading the wrong books, but who can say because
you do not tell us what they are.

R.G. Vickson


Very much thanks in advance,
Regards,
Sujit P Gujar.
IISc Bangalore.
Web:http://people.csa.iisc.ernet.in/sujit

.

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