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Title:	On the ""fair`` games problem for the weighted generalized Petersburg games
Authors:	林光賢 Lin, Kuang Hsien 陳天進 Chen, Ten Ging Yang, Ling-Huey
Contributors:	應數系
Date:	1993-03
Issue Date:	2018-09-25 16:22:01 (UTC+8)
Abstract:	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``.
Relation:	Chinese Journal of Mathematics,21(1),21-31 AMS MathSciNet:MR1209488
Data Type:	article
Appears in Collections:	[應用數學系] 期刊論文

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