Re: Basic Doubts regarding sequence of Random Variables and Stochastic

Sujit <sujit.gujar@xxxxxxxxx> writes:

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?

They are the same.

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.

Yes.

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.

They are not independent.

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

Check the definition of independent.

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?

Either.

X_i are all functions on the sample space, so they depend on the outcome.
But they don't have to depend on the whole outcome. This leads to a
discussion of sub-sigma-fields and filtrations.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Re: Basic Doubts regarding sequence of Random Variables and Stochastic Process?
    ... I have couple of basic doubts regarding sequence of random variables/ ... discrete time stochastic process defined ... A = set of all sequence of Random Variables on some probability ...
    (sci.math)
  • Basic Doubts regarding sequence of Random Variables and Stochastic Process?
    ... I have couple of basic doubts regarding sequence of random variables/ ... discrete time stochastic process defined ... A = set of all sequence of Random Variables on some probability ... So say, coin toss, and if head on ith coin toss, X_i = 1 else 0. ...
    (sci.math)
  • Re: How Chapman-Kolmogorov implies Markov ??
    ... Perhaps the notation "f_i" is supposed to imply there is ... something equivalent about all of these random variables, ... concept, that we index the index set iself, or else write a finite ... sequence in it, explicitly using the integers as our indices. ...
    (sci.physics)
  • Re: How Chapman-Kolmogorov implies Markov ??
    ... something equivalent about all of these random variables, ... concept, that we index the index set iself, or else write a finite ... sequence in it, explicitly using the integers as our indices. ... engineer his intentions through the ambiguities, ...
    (sci.physics)
  • Re: Good enough for crypto?
    ... > random variables first as being deterministic, ... >> Your iid sequence is a mathematical abstraction. ... It is only the measurements that are probabilistically distributed, ...
    (sci.crypt)