00266 Ù4691a816ffaef1f9b3e8635d95fa34

268

Yander Wiel

Arguments for Theorems 2 and 3 For clarity we set Y_i = |Z,|. To make the

arguments elear, we take the liberty of letting the symbol Pr represent a probability density as well as the probability of an event.

To see why Theorem 2 follows from Theorem 1 consider the following:

Pr{maxT_A = u) = ^Pr{l* = «}Pr{all other Ysś. u\Y_k = u}

k

= Pr{T₀ = t/}^Pr{ prior to k, all Fs^t/IT* =u}

k

xPr{afterA:, aUTs<«|y_A =u}

The second equality follows from the first because the Ys are identically distributed (with density 2<j>(') on (0,oo)) and given Y_k the Ts prior to k are independent of the Ys following k due to the Markov property.

To see why the recursion for F_k is possible notice that g*(-|v) is the density given that Y_k = v and then consider the following:

^(wly) = Pr{prior to k, all Xs <    =v}

J^v _q Pr{prior to A- — 1, all ł^s < w|T^rA_, = s, Y_k = v}g*(s)v)*/s

The second eÄ…uality results from conditioning on y*_, in addition to Y_k. To get the third equality Y_k is dropped as a conditioning variable. This is valid because the Markov property implies that (Y_Q,...,Y_k_₂) and Y_k are independent given Y_k__x. The argument giving the recursion for F_k is similar.