Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 3, 884--893

SOME EXTENSIONS OF THE SKOROHOD REPRESENTATION THEOREM

By

JAYARAM SETHURAMAN, Florida State University, USA

SUMMARY. Let $\{\mu_n\}_{n=0}^{\infty}$ be a sequence of probability measures on a measurable space ({\mathcal X},{\mathcal A})$. The famous Skorohod representation theorem Skorohod~(1956) says that when$\mu_n$converges to$\mu_0$weakly, we can find random variables$\{X_n,n=0,1,\dots\}$distributed marginally as$\{\mu_n,n=0,1,\dots\}$, which converge almost surely to$X_0$. If the hypothesis is strengthened to$||\mu_n - \mu_0||\r 0$, where$||\cdot||$is the variation norm for measures, one can get the stronger conclusion that the above random variables$X_n$can be chosen so that$P(X_n = X_0) \r 1$. This result is a corollary of available results in the literature; see e.g. Dobrushin~(1970). In this paper we show that if we assume a still stronger condition, namely that the pdf's$f_n$of$\mu_n$w.r.t a measure$\nu$, satisfy$\liminf_n f_n(x) \ge f_0(x)$a.e$[\nu]$where$f_0$is the pdf of$\mu_0$, then the random variables above can be chosen so that$P(X_n = X_0$ultimately$) = 1$. We also show that these results are tight by showing that the conditions are necessary and sufficient. We conclude the paper with illustrative examples and show that$\limsup_n f_n(x) = f_0(x)$may not imply$\mu_n \r \mu_0\$ in any sense.

AMS (1991) subject classification}. 60B10, 60F15.

Key words and phrases. Skorohod representation theorem, Vasershtein metric, Scheff\'e's theorem.

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