Definition: Let X(adsbygoogle = window.adsbygoogle || []).push({}); _{1},X_{2},... be a sequence of random variables defined on a sample space S. We say that X_{n}converges to a random variable X in probability if for each ε>0, P(|X_{n}-X|≥ε)->0 as n->∞.

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Now I don't really understand the meaning of |X_{n}-X| used in the definition. Is |.| here the usual absolute value when we talk aboutreal numbers? But X_{n}and X arefunctions, not real numbers.

Also, when we talk about the probability of something, that something has to be subsets of the sample space Ω, but |X_{n}-X|≥ε does not look like the description of a subset of Ω to me.

A random variable X is a function mapping the sample space Ω to the set of real numbers, i.e. X: Ω->R.

The random variable X is the function itself, and X(ω) are the VALUES of the function which are real numbers. Only when we talk about X(ω) does it make sense to talk about the absolute value of real nubmers.

So I assume the notation used in the definition above is really a shorthand for

P({ω E Ω: |X_{n}(ω)-X(ω)|≥ε})?

In other words, the notations P(|X_{n}-X|≥ε) and P({ω E Ω: |X_{n}(ω)-X(ω)|≥ε}) are interchangable? Am I right?

I hope someone can clarify this! Thanks a lot!

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# Random Variables: Convergence in Probability?

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