Basic Doubts regarding sequence of Random Variables and Stochastic Process?

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?


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

Does X_n converge to 0 almost surely(a.s). 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.
So only thing can go wrong is X_n are not independent, which is
also required to apply BC lemma.

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


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,

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

.

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