有關chow-robbins的\"公正\"遊戲問題之探討

Abstract

令Sn=Σj =1najYj ，其{Yn,n≧1}是具有相同分布的獨立隨機變數序列，且{an , n≧1}為正值實數數列。考慮一連串的比賽遊戲，以anYn表示參與者於第n次比賽時，所獲得的”利益”;且假設欲參與第n次比賽遊戲時，須預先支付賭注mn。在本文中，我們證明:若比賽遊戲採用的是”Generalized Petersburg Games”,即p{Y1=q-k}=pqk-1,0<p=1-q<1,k≧1;且若正值實數數列{an,n≧1}滿足

Let Sn=?_(j=1)^( n)??a_j Y_j ?, n≧1,where{Yn, n≧1}are i.i.d. r.v.’s and{an,n≧1}are real numbers. Interpreting an Yn as a player’s winnings from the n-th game,a natural question is whether there is an entrance fee mn to the n-th game such that Sn / Mn → 1 in pr. where Mn= ?_(j=1)^( n)?mj.The Purpose of this paper is to study a generalization of the classical Petersburg game for the weighted i..i.d case. That is, for a sequence{ an,n≧1} of real numbers and i.i.d.r.v.’s { Yn, n≧1}with P{ Y1=q-k}=pqk-1, 0<p=1-q<1, k≧1,find conditions on {an,n≧1}which ensure the existence of constants {Mn, n≧1} for which Sn / Mn－1 in pr. obtains. It is shown that when an≧0, An=1,2,3,.....