ec1f764517b7ffb52057af6df18142b7-Supplemental.pdf

Theeventsintasksisdenoted as Hs = Hs,n+1 and all the events upto and including stages is denoted asH1:s = ss0=1Hs0. Let X denote that the proportionality constant isindependentofX (possiblyaset). Using the relation between the history-independent and history-dependent entropy terms we obtain the second inequality. The right side term in the first inequality holds due to Cauchy-Schwarz. The inequality holds due to Weyl's inequality and This is true by using the upper bounds onσ2max(ˆΣs,t) in Lemma 3, and because the functionp x/log(1+ax)fora>0increaseswithx.