Suppose $Y_n:=v_n^{-1}\sum_{i=1}^n X_i\xrightarrow{}Z\sim N(0,1)$ in distribution where $(X_n)_{n\in\mathbb{N}}$ is a stationary sequence of real random variables with finite variance and $v_l$ is a appropriate standardisation.

Is it possible to assume that there exists a random variable $\tilde{Z}$ with $Y_n\xrightarrow{}\tilde{Z}\sim N(0,1)$ pointwise? Or do we have to change the probability spaces where $Y_n$ and $Z$ originate from as in Skorohod's representation theorem?


  • 2
    $\begingroup$ In general you cannot. Since you can have a sequence that converges in distribution to some law and not converging in probability to any random variable. Thus you cannot have almost sure convergence of the sequence. $\endgroup$ – Bunder Oct 2 '13 at 9:57
  • $\begingroup$ @Bunder: Thanks a lot! $\endgroup$ – stroem Oct 4 '13 at 7:37

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