  A. Vershik discovered that filtrations indexed by the non-positive integers
may have a paradoxical asymptotic behaviour near the time $-\infty$, called
non-standardness. For example, two dyadic filtrations with trivial tail
$\sigma$-field are not necessarily isomorphic. Yet, any essentially separable
filtration indexed by the non-positive integers becomes standard when
sufficiently many integers are skipped. In this paper, we focus on the non
standard filtrations which become standard if (and only if) infinitely many
integers are skipped. We call them filtrations at the threshold of
standardness, since they are as close to standardardness as they can be
although they are non-standard. Two class of filtrations are studied, first the
filtrations of the split-words processes, second some filtrations inspired by
an unpublished example of B. Tsirelson. They provide examples which disproves
some naive intuitions. For example, it is possible to have a standard
filtration extracted from a non-standard one with no intermediate (for
extraction) filtration at the threshold of standardness. It is also possible to
have a filtration which provides a standard filtration on the even times but a
non-standard filtration on the odd times.
