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.
For the derivativesp(k)(x; α, γ) of the stable density of index α asymptotic formulae (of Plancherel Rotach type) are computed ask→∞ thereby exhibiting the detailed analytic structure for large orders of derivatives. Generalizing known results for the
special case of the one-sided stable laws (O<α<1, γ=-α) the whole range for the index of stability and the asymmetry parameter γ is covered.
is thoroughly studied, comparing and completing the known results obtained through the three different approaches mentioned
in the title. A unifying thread is provided by the canonical factorizations of the differential operator Dn. Particularly meaningful are several characterizations of the polynomial asymptotic expansions of an nth order convex function.
The asymptotic behavior of the values of the integral of the Lebesgue function induced by interpolation at the Chebyshev roots
is studied. Two leading terms in the corresponding asymptotic expansion are found explicitly.
A collector samples with replacement a set of
≧ 2 distinct coupons until he has
, 0 ≦
, distinct coupons for the first time. We refine the limit theorems concerning the standardized random number of necessary draws if
→ ∞ and
is fixed: we give a one-term asymptotic expansion of the distribution function in question, providing a better approximation of it, than the one given by the limiting distribution function, and proving in particular that the rate of convergence in these limiting theorems is of order (log