Limits and Inferences for Alpha–Stable Variables

Distributions having excessive tails are modeled in various venues via α–stable sequences deficit in moments of various orders. Essential statistics are examined under independent vs spherically dependent cases. The former exhibits such critical pathologies as inconsistent sample means. The latter support versions of some classical procedures even without moments as conventionally required. In particular, despite heavy tails, Student’s tests for means nonetheless remain exact in level and power. etc. AMS Subject Classification: 62E15, 62H15, 62J20.

Excessive tails typically are modeled through sequences. An outline follows. Notation and technical foundations are provided in Section 2. The findings in Section 3 are twofold: First, that limit properties diverge widely between and ( ) iid spherically dependent ( ) S S α sequences; and second, that conventional inferences, though largely lacking in the former, may be validated in large part in the latter. Conclusions are tallied in Section 4.

Special distributions
µ Σ Distributions on 1  include the ( ) 2 ; , u v χ λ having degrees of freedom and non centrality parameter ; λ and the corresponding Student's ( a standard source is Lukacs & Laha [11]. Reference is drawn subsequently to probability density (PD) and cumulative distribution (CD) functions.

Foundations
for some function ( ) . ψ not depending on N [12,13]. Averages and limits are basic in statistical analysis; for example, notions of consistency in estimation and of large-sample distributions. These are undertaken here without benefit of moments in keeping with excessive tails. Specifically, the sequence { } respectively. In addition, the following is central to subsequent developments. Then there chfs and pdfs are related as follows. .
is a scale mixture of N-dimensional spherical Gaussian chfs. This applies in context to give conclusion (i). To continue, is the standard inversion formula from chfs to densities with ( ) Λ ⋅ as Lebesgue measure. Accordingly, we invert both sides of the second and third expressions in conclusion (i) to get the density on the left of conclusion (ii). We then recover the right side of conclusion (ii) on reversing the order of integration in the iterated integral found on inverting the third expression of conclusion (i). might also be independent. To the contrary, for any spherical sequence. Maxwell [14] showed this to be the case if and only if Gaussian. In view of this, it remains to examine the limit properties of iid S S α Variables in comparison with spherically dependent S S α Variables in N  of critical interest to users. Limit properties of these are shown next to be widely disparate, despite the fact that their marginals coincide. At issue are statistics

Sequences of iid and S S
in order to be iid. A principal finding follows.

I. Theorem 1
Take elements of [ ] which, by the Levy-Cramer continuity theorem, is the chfs of a distribution degenerate at .
δ Consistency of N Z for then follows using the equivalence of convergence in law to degeneracy, and convergence in probability.

II. Theorem 2
Limit properties of N Z and ( )   α < ≤ On the other hand, the next section motivates circumstances for the occurrence of spherical S S α samples, and sets out to establish useful statistical properties in the analysis of data from these models.

Properties of spherical S S α samples
Consider anew a single sample with parameters ( ) The following is central to our findings.

Conclusion
In practice scale mixtures may arise as conditionally iid Gaussian variables subject to scaling in a random environment. Linear models so structured are treated in Zellner [15]