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| Title: | 有關chow-robbins的"公正"遊戲問題之探討
ON THE CHOW-ROBINS "FAIR" GAMES PROBLEM |
| Authors: | 楊玲惠 |
| Contributors: | 林光賢
楊玲惠 |
| Date: | 1990
1989 |
| Issue Date: | 2016-05-03 14:17:35 (UTC+8) |
| 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,..... |
| Reference: | [ 1] A. Adler and A. Rosalsky , On the Chow-Robbins " fair “ games problem, Bulletin of the institute of mathematics academia sinica . , 17 (1989) ) 211-227
[ 2 ] Y. S. Chow and H. Robbins, On sums of independent random variables with Infinite moments and “fair “ games) Proc. Nat. Acad. Sci. U.S.A.,47(1961) , 330-.335 .
[ 3 ] Y. S. Chow and H. Teicher , Probability Theory : Independence , Interchangeability , Mrartingale , Springer-Verlag, New York, 1988 .
[ 4] W. Feller 1 Note on the law of large numbers and “ fair" games, Ann.Math. Statist. , 16 (1945) , 301-304 .
[ 5] W. Feller. , A limit theorem for random variables with infinite moments, Amer. J. Math. , 68 (1946) ,257-262 .
[ 6] W. Feller. , An Intruductin to Probability Theory and Its Applications, Vol I, 3rded. , John Wiley, New York, 1968 .
[ 7 ] W. Feller. , An Intruductin to Probability Theory and Its Applications, Vol II, 2nded. , John Wiley, New York, 1971 .
[ 8] B. Jamison, S. Orey and W. Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete ,14 (1965) , 40--44 .
[ 9] R. A. Maller, Relative stability and the strong law of large numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , 43 (1978) , 141-148 .
[10J B. A. Rogozin , Relatively stable Walks , Theor. Probability Appl. , 21(1976) ,375--379 . |
| Description: | 碩士
國立政治大學
應用數學系 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#B2002005452 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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