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Domains of attraction and regular variation in \({\mathbb{R}}^ d\). (English) Zbl 0536.60027

The authors define an extended form of regular variation of functions on \({\mathbb{R}}^ 2_+\). The function f: \({\mathbb{R}}^ 2_+\to {\mathbb{R}}_+\) varies regularly at infinity with auxiliary functions r: \({\mathbb{R}}_+\to {\mathbb{R}}_+\) and s: \({\mathbb{R}}_+\to {\mathbb{R}}_+\) if \(f(xr(t),ys(t))/f(r(t),s(t))\to \lambda(x,y), x>0\), \(y>0\), as \(t\to \infty\). For f monotone in each variable and r,s both varying regularly at \(\infty\), they prove the extensions to \({\mathbb{R}}^ 2_+\) of the well known theorems for regular variation in \({\mathbb{R}}_+:\) Karamata’s integral theorem, regular variation of a monotone derivative of f and the Abel-Tauber theorem connecting the behaviour of f at infinity and of the Laplace transform of f at zero.
These results are applied to the stable attraction of random vectors in \({\mathbb{R}}^ 2:\) Let \((X_ i,Y_ i)\) be i.i.d. random vectors. If \((a_ n^{-1}\sum^{n}_{k=1}X_ k-c_ n\), \(b_ n^{- 1}\sum^{n}_{k=1}Y_ k-d_ n)\) converges in distribution to nondegenerate (V,W), the distribution of (V,W) has to be stable. The possible stable limit distributions and conditions for convergence were determined by S. Resnick and P. Greenwood, ibid. 9, 206-221 (1979; Zbl 0409.62038). For \(X_ i>0\), \(Y_ i>0\) the authors give new proofs of some of these results and also new conditions for convergence in terms of regular variation of \(P(X_ 1>x\), \(Y_ 1>y)\) and its integral. Finally, they succeed in transferring the convergence theorems to not necessarily positive \(X_ i\) and \(Y_ i\), if (V,W) or V is normal.
Reviewer: A.J.Stam

MSC:

60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
26A12 Rate of growth of functions, orders of infinity, slowly varying functions

Citations:

Zbl 0409.62038
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References:

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