Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations
of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any
one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms
and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic
distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For
all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that
for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic
expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be
numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations
even for very small n.
Based on a stochastic extension of Karamata’s theory of slowly varying functions, necessary and sufficient conditions are
established for weak laws of large numbers for arbitrary linear combinations of independent and identically distributed nonnegative
random variables. The class of applicable distributions, herein described, extends beyond that for sample means, but even
for sample means our theory offers new results concerning the characterization of explicit norming sequences. The general
form of the latter characterization for linear combinations also yields a surprising new result in the theory of slow variation.
Summary General linear combinations of independent winnings in generalized \St~Petersburg games are interpreted as individual gains that result from pooling strategies of different cooperative players. A weak law of large numbers is proved for all such combinations, along with some almost sure results for the smallest and largest accumulation points, and a considerable body of earlier literature is fitted into this cooperative framework. Corresponding weak laws are also established, both conditionally and unconditionally, for random pooling strategies.
Authors:Vl. Simeon, B. Momčilović, Z. Kralj, and B. Grgas
The publications produced in a medical research institute in a 16 year interval were classified into five categories (scientific papers in the journals covered byCurrent Contents orScience Citation Index, scientific papers in other journals, books and monographs, technical papers, congress and symposia communications) and counted for each year separately. The number of researchers and yearly budgets were also recorded. The data were analysed by contingency table, correlation and factor-analytical methods. It was shown that, upon introducing quantitative minimal criteria for job promotions, the proportion of scientific papers increased. Principal component analysis indicated that the data can be approximately represented as linear combinations of three mutually independent factors. The approach used is recommended for evaluating the production of scientific information in research institutions and for assessing the effects of the measures of scientific policy.