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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 $n, x$.) –  Nate Eldredge Oct 31 '12 at 12:50
    
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). –  Did Oct 31 '12 at 18:35

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@ Nate :In the special case where $Z_n$ ≡ $z_n$ with constants $z_n$ (n=1,2,...) the definition coincides with the usual $\liminf _{n\to \infty}Z_n$, otherwise not. The definition has just come from the book by Te Sun Han : Information Spectrum Methods in Information Theory.(p14,def 1.3.2).Ofcourse the definition is somewhat vague if we have non-stationary source,i.e. $Z_i$~p($z_i$) for $z_i$$\in$ $\{\beta_i\}$. Suppose the cdf of $Z_i$ ($Pr\{ Z_i <\beta \}$) is zero for $\beta=\beta_{Z_i}$ then we can $\sup$ the $\beta_{Z_i} $ ! By the way,the indicator function you just defined is not clear to me in argument !

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