| dc.description.abstract (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 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``. | en_US |
