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Title: On the ``fair'' games problem for the weighted generalized Petersburg games
Authors: 林光賢
Lin, Kuang Hsien
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 &quot;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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