= Time-zero correction for a half-point dyadic spike martingale
{title2=$X_n=2^n\mathbf1_{(1/2-2^{-n},1/2]},\quad n\geq1$}
On $(0,1]$ with Lebesgue probability and the usual <dyadic filtration>, each displayed support is a dyadic atom. Its next split has one child of doubled height and one of zero height, preserving the conditional average. Hence the sequence is a <martingale> from time one. To extend it to the trivial time-zero <sigma-algebra> one must set $X_0=1$: applying the same displayed formula at zero instead gives $\mathbf1_{(0,1/2]}$, which is not adapted and has the wrong <expected value>. The repaired <martingale> tends to zero almost surely while its <expected values> remain one, so it is not uniformly integrable.
Back to article page
