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"$X_n$'s are bounded in probability" means "for every $\epsilon >0$ $\exists M_{\epsilon}$ such that $P(|X_n|>M_{\epsilon})<\epsilon$ $\forall n\geq1$ i.e. a single $M_{\epsilon}$ will work for all $n$. Now does that mean $X_n$ must be defined on the same probability space(like a.s./in probability convergence)? The same question arises(at least from my mind) when we say $\{X_n\}$ is uniformly integrable.

As when we say $X_n$ converges in distri. to $X$ all I am concerned about is their distri. function $\{F_n\}$. Thus if I know $F_n$ I can figure out whether $X_n=O_p(1)$ or not. Thus can I replace $P$ by $P_n$ in the definition of bounded in probability similarly in uniformly integrable?

As I've just started learning probability theory my concept is not quite clear regarding these. Would you please clarify my query/conceptual understanding? Thank you,

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You are correct -- the concept of boundedness in probability does not require all the $X_n$'s to be defined on the same probability space, because the concept concerns the distribution of each $X_n$. Similarly the concept of uniform integrability makes sense even if the $X_n$'s live on different spaces.

Only concepts that refer to pointwise behavior of the $X_n$'s (such as the sum of random variables, or the limsup of a sequence of RVs) absolutely require everything to be defined on the same space.

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  • $\begingroup$ Or simply the almost sure convergence of the sequence $(X_n)$. $\endgroup$
    – Did
    Aug 14 '15 at 17:55
  • $\begingroup$ @grand_chat Thank you! $\endgroup$
    – Robert
    Aug 14 '15 at 19:44

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