What is the connection between topology and convergence in probability?

[wayneckm]

Hello all,

Recently I come across the description of "a space topologized by convergence in probability". And I can search that it is the topology generated by a metric defined by $$E\left[ \frac{|X|}{1+|X|}\right]$$.

I can understand how a topology related to a metric. However, whenever we come across the word "topologize by some convergence", does that implicitly mean that we should work out the corresponding metric defined for such convergence and then generate the topology by using this metric?

Lastly, can anyone explain why there is no topology associated with a.s. convergence since I cannot find any reference regarding this? Does this mean there is no well-defined metric for a.s. convergence?

Thanks.

 

[lavinia]

could you explain this a little more. What is X? some random variable?

 

[g_edgar]

Yes, take distance between random variables $X$ and $Y$ to be
$$E\left[\frac{|X-Y|}{1+|X-Y|}\right]$$
So... convergence in this metric is the same as convergence in probability. Then go to the topology defined by the metric. That's what they mean "topologized by convergence in probability". Presumably they wanted to use some topological term, so they needed to say what topology was to be used.

Second: You are correct. There is no metric (and indeed no topology) whose convergence coincides with convergence a.e.

 

[lavinia]

Is this metric space complete? If not then you would get a new topological space where the points are equivalence classes of Cauchy sequences. What would this be like? Would it be a space of random variables?

 

[wayneckm]

Thanks for all replies.

So another question comes to my mind is:

Whenever one said a space topologized by some convergence mode, it apparently that one always boils down to struggle and work out the suitable metrics in order to characterize the open set in such topology and so one can continuous analysis with them. However, what if one fails to figure out the corresponding metric, so is it still possible to analyze without the explicit structure of topology endowed with this convergence mode?

Or is there any other way to talk about open set under some convergence mode without relying on metric?

People saying "topologized by some convergence mode" apparently indicate me that they primiarily use the notion of convergence to characterize open set, while metric compatible to such topology can luckily be found in some cases (hence turning such topological space into metric space), so there can be some siutations that we cannot do so and so what to do with them?

Thanks.

 

[Bacle]

wayneckm:

I wonder if this may help:

http://planetmath.org/encyclopedia/TopologyViaConvergingNets.html

It is a little late in the day for my presenting it.