ABSTRACT
Assuming that we have made an appropriate choice of the sets { A11 } satisfying (i) and (ii), we need to obtain a bound on the difference
IIP*(T~:::; x)- ~11 (x)llx on the set A11 for 11 ~ 110 • To do this, we will first derive an Edgeworth expansion for T~ upto the desired degree of accuracy, by further conditioning with respect to the block length variables L 1, ••• , L 11 , and then integrate the resulting expression (with respect to the joint distribution of L 1 , ••• , L11 given X 1 , ••• , X11 ) on A11 to arrive at inequality (5.1). Recall that P** (and£**) denotes the conditional probability (and expectation,respectively) given X 1,X2, •.• and {L 1, ••• ,L,,} 11 > 1• Hence, it follows that for any .1" -measurable random variable Y, -
so that
sup jP*(Ti,:::; x)- E*{P**(Ti,:::; x)IA,~}I14 ,:::; P*(A~'") · lA,· (5·3) YE~
l :::; i:::; n,
for all I :::; j -::; n. Hence, using properties of the circularly extended sequence {X11;: i ~ l} one can show (cf. Politis and Romano (1994)), that
suitable stochastic expansion for T~ and then use the transformation technique of Bhattacharya and Ghosh (1978) and the Edgeworth expansion theory for sums of independent random vectors (cf. Bhattacharya and Ranga Rao (1986)) to the resulting stochastic expansion. Without loss of
