Re: Question in Understanding "converges in probability"
- From: "Nasser Abbasi" <nma@xxxxxxxxx>
- Date: Mon, 17 Dec 2007 05:00:41 -0800
"Tim" <liting0612@xxxxxxxxx> wrote in message
news:7a7e5925-4667-41bf-8a15-83c0a257646d@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
"The definition says "{Xn} converges in probability to X iff P(|Xn - X|>=
å) ---> 0 for every å > 0". I am wondering if it can be defined
without introducing å? Specifically, what's the difference between "P(|Xn -
X| > å) ---> 0 for every å > 0" and "P(|Xn - X| > 0) ---> 0"? If they are
not the same, as a condition which one is stronger ? Please give some
clarification. Thanks in advance!"
I am not sure myself. This definition was always a bit not clear to me, but
let me give it a shot.
First, the definition in words is, I think, as follows:
Imagine you have some process which generates one random variable after
another. Imagine also you have some fixed scalar value. For illustration,
lets say X is some point in space, which is the limit we are looking at in
this case. Imagine this point X being the center of a circle of radius
epsilon where one can make the radius as large or small as we wish.
The definition then says: as more and more r.v.'s are generated, eventually
the *probability* that the next one that comes out would be further away
from X by some epsilon amount would tend to zero, regardless of what radius
one picks.
In other words, the probability that, in the limit, that a random variable
will fall outside the circle centered at X will approach zero, no matter how
small the radius of the circle is. i.e. eventually the probability that the
nth r.v. variable, for very large n, will land inside the circle approaches
a certainty for any radius you pick no matter how small.
Now lets look at your alternative definition. You are saying in words: As n
gets very large, The probability that the nth r.v. will not land almost at
the center of the circle will tend to be zero.
Humm. Well, your definition seems ok, in the limit, but it seems to not show
the convergence process. i.e. You are just saying that the probability that
the sequence limit is different from its limit gets to be zero but not shown
a mechanism of how this limit progresses. I think this epsilon is important
for this part. (it is also important to make student work harder :)
Also notice the definition say for any 'epsilon'. Your definition does not
have epsilon at all, so one can't use your definition to say something as
follows: the limit of the rv. sequence, will have a probability of it being
away from its X by more than 5 is approaching zero. Where I just picked 5,
because of the 'for any epsilon' in the definition, I am allowed to do so.
right?
In your definition, I could only say: the limit of the rv. sequence have a
probability approaching zero of being different from its limit X.
Any way. this is my 2 cents attempt at this late at night. I think this is a
good question. You might want to look at convergence almost surely
definition, it is a stronger convergence than in probability and might help.
I think if I take another course in statistics, we might study this stuff in
more details.
Nasser
.
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