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請使用永久網址來引用或連結此文件:
https://nccur.lib.nccu.edu.tw/handle/140.119/120128
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| 題名: | On the ""fair`` games problem for the weighted generalized Petersburg games |
| 作者: | 林光賢
Lin, Kuang Hsien
陳天進
Chen, Ten Ging
Yang, Ling-Huey |
| 貢獻者: | 應數系 |
| 日期: | 1993-03 |
| 上傳時間: | 2018-09-25 16:22:01 (UTC+8) |
| 摘要: | Let $S_n=\\sum^n_{j=1}a_jY_j$, $n\\geq 1$, where $\\{Y_n,\\ n\\geq 1\\}$ is a sequence of i.i.d. random variables with the generalized Petersburg distribution $P\\{Y_1=q^{-k}\\}=pq^{k-1}$, $k\\geq 1$, where $0<p=1-q<1$ and $a_n,\\ n\\geq 1$, are positive constants with $(\\sum^n_{j=1}a_j)/\\max_{1\\leq j\\leq n}a_j\\to\\infty$. The main result asserts that $S_n/M_n\\overset P\\to\\rightarrow 1$, where $$M_n=\\sup\\Big\\{x\\colon\\ \\sum^n_{j=1}a_jEY_1I(a_jY_1\\leq x)\\geq x\\Big\\},\\quad n\\geq 1,$$
thereby generalizing a result of A. Adler and the reviewer [Bull. Inst. Math. Acad. Sinica 17 (1989), no. 3, 211–227; MR1042179] obtained for the particular choice $a_n=n^\\alpha$, $n\\geq 1$, where $\\alpha>-1$. This problem has the following interesting interpretation. Suppose a player wins $a_nY_n$ dollars during the $n$th game in a sequence of generalized Petersburg games. If $M_n=\\sum^n_{j=1}m_j$ represents the accumulated entrance fees for playing the first $n$ games, then $S_n/M_n\\overset P\\to\\rightarrow 1$ is the assertation that $\\{m_n,\\ n\\geq 1\\}$ is a "fair solution in the weak sense to the games``. |
| 關聯: | Chinese Journal of Mathematics,21(1),21-31
AMS MathSciNet:MR1209488 |
| 資料類型: | article |
| 顯示於類別: | [應用數學系] 期刊論文
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