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For an arbitrary sequence of real-valued random variables $\{Z_n\}_1^\infty$ , we define limit inferior in probability as follow : $$ p-\liminf_{n\to \infty} Z_n \equiv \sup \{ \beta|\lim_{n\to \infty} Pr\{ Z_n <\beta \}=0\} $$Can any one elaborate this limit operation via an example? Please introduce some references about this question in mathematics context.

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    I don't think this can be the right definition. Assuming $\beta$ is a random variable and the supremum is taken pointwise, you can have the situation that $Z_n = 0$ for every $n$ but $\liminf Z_n = +\infty$. (For example, if your probability space is $[0,1]$ with Lebesgue measure, you can take $\beta = n 1_{\{x\}}$ for any $n, x$.)2012-10-31
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    You could make precise what $\beta$ is, a real number or a random variable (the notion of liminf one gets in each case are quite different).2012-10-31

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