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.