On some Differences in Limit Theorems with a Normal or a Non‐normal Stable Limit Law

Abstract

Suppose that the distribution function of a standardized sum of independent identically distributed random variables tends to a stable law as n + co. Some differences in moments and pseudomoments, inequalities of the BERRY-ESSEEN-Type and asymptotic expansions are characterized when the limit law is either normal or non-normal stable.

Some remarks on the rate of convergence to stable laws

“…Now we choose Cz and Cz coinciding with the first two coefficients in the expansion of p,,,# in (2), although it is clear that the value of C2 is important only in the case y = 2~. Using Theorem 4.12 in [3] and Theorem 1 in [7] or by direct calculations, one can derive the following bounds.…”

“…What properties of the summands have the most influence on the rate of convergence under consideration? At once it is necessary to note that a general answer is well known at present (a good reference for the accuracy of approximation with stable laws is the recent monograph [3], see also [6]- [11]), and it can be roughly stated that the existence of pseudomoments of higher order ensures better rates of convergence. In turn, the existence of pseudomoments depends on how close are the distribution of the summand and the stable law.…”

Theory of a Systematic Computational Error in Free Energy Differences

“…More precisely, the sum of any set of random variables, suitably normalized as in (7), has a distribution with zero mean which may be expressed as a stable distribution function multiplied by a large-n asymptotic expansion [14,19].…”

On asymptotic properties of multidimensional α ‐stable densities

“…Analytical properties of univariate stable distributions are well studied (see e.g. [31] and [5,10,11,29]). …”