Let Pn and Qn be two sequences of probability measures defined on (n, Fn). Assume they are

Let Pn and Qn be two sequences of probability measures defined on (Ωn, Fn). Assume they are contiguous. Assume further that both of them are product measures, i.e.

Pn = n i=1 Pn,i and Qn = n i=1 Qn,i .

Let Q − P1 denote the total variation distance between P and Q. Show that sup n

n i=1

Qn,i − Pn,i2 1 < ∞ .