Re: Question in Understanding "converges in probability"
- From: "David Jones" <dajxxxx@xxxxxxxxx>
- Date: Mon, 17 Dec 2007 13:41:04 -0000
Tim wrote:
The definition says "{Xn} converges in probability to X iff P(|Xn - X|
= ε) ---> 0 for every ε > 0". I am wondering if it can be definedwithout introducing ε? Specifically, what's the difference between
"P(| Xn - X| > ε) ---> 0 for every ε > 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!
Things way be a liitle clearer if you remember that for, a continuous random variable X,
Pr(X=0)=0, hence
Pr(|X|=0)=0 also,
so Pr(|X|>0)=1 always.
So P(|Xn - X| > 0) ---> 0, only if (eventually) |Xn - X| has a discrete component at zero, and this component has a probability approaching 1 as n increases.
David Jones
.
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