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.
Moment inqualities and strong laws of large numbers are proved for random allocations of balls into boxes. Random broken lines and random step lines are constructed using partial sums of i.i.d. random variables that are modified by random allocations. Functional limit theorems for such random processes are obtained.
Authors:István Berkes, Wolfgang Müller, and Michel Weber
Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X1,X2, … is any sequence of integrable i.i.d. random variables, then
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.
As the variance (the square of the minimum L 2-norm, i.e., the square of the scatter) is one of the basic characteristics of the conventional statistics, it is of practical importance to know the errors of its determination for different parent distribution types. This statement is outstandingly valid for the geostatistics because the (h) variogram (called also as semi-variogram) is defined as the half variance of some quantity-difference (e.g. difference of ore concentrations) in function of the h dis- tance of the measuring points and this g (h)-curve plays a basic role in the classical geostatistics. If the scatter (s VAR) is chosen to characterize the determination uncertainties of the variance (denoted the latter by VAR), this can be easily calculate as the quotient A VAR= Ön (if the number n of the elements in the sample is large enough); for the so-called asymptotic scatter A VAR is known a simple formula (containing the fourth moment). The present paper shows that the AVAR has finite value unfortunately only for about a quarter of distribution types occurring in the earth sciences, it must be especially accentuate that A VARhas infinite value for that distribution type which most frequent occurs in the geostatistics. It is proven by the present paper that the law of large numbers is always fulfilled (i.e., the error always decreases if n increases) for the error-determinations if the semi-intersextile range is accepted (instead of the scatter); the single (quite natural) condition is the existence of the theoretical variance for the parent distribution. __
The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (Xk: k = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN
which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The
results are also extended for random fields (Xkℓ: k, ℓ = 1, 2, …).