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''domain of attraction of a stable distribution''
 
''domain of attraction of a stable distribution''
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{{MSC|60F05|60E07}}
  
 
[[Category:Limit theorems]]
 
[[Category:Limit theorems]]
 
[[Category:Distribution theory]]
 
[[Category:Distribution theory]]
  
{{User:Rehmann/sandbox/MSC|60F05|60E07}}
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The totality of all distribution functions $F(x)$ such that for a sequence of independent identically-distributed random variables $X_1,X_2,\dots,$ with distribution function $F(x)$ and for a suitable choice of constants $A_n$ and $B_n>0$, $n=1,2,\dots,$ the distribution of the random variable
 
 
The totality of all distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139201.png" /> such that for a sequence of independent identically-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139202.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139203.png" /> and for a suitable choice of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139206.png" /> the distribution of the random variable
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\frac{\sum_{k=1}^nX_k-A_n}{B_n},\qquad n=1,2,\dots,\label{*}\tag{*}$$
  
converges weakly, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139208.png" />, to a non-degenerate distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139209.png" />, which is necessarily stable.
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converges weakly, as $n\to\infty$, to a non-degenerate distribution function $V(x)$, which is necessarily stable.
  
One of the fundamental problems in the theory of stable laws is the description of domains of attraction of stable laws. Thus, for the normal distribution, A.Ya. Khinchin, W. Feller and P. Lévy established in 1935 that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392010.png" /> belongs to the domain of attraction of a normal law if and only if, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392011.png" />,
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One of the fundamental problems in the theory of stable laws is the description of domains of attraction of stable laws. Thus, for the normal distribution, A.Ya. Khinchin, W. Feller and P. Lévy established in 1935 that $F(x)$ belongs to the domain of attraction of a normal law if and only if, as $x\to\infty$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392012.png" /></td> </tr></table>
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$$x^2\frac{\int\limits_{|y|>x}dF(y)}{\int\limits_{|y|<x}y^2\,dF(y)}\to0.$$
  
Later B.V. Gnedenko and W. Doeblin (1940) gave a description of the domain of attraction of a stable law with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392014.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392015.png" /> belongs to the domain of attraction of a non-degenerate stable law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392016.png" /> with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392017.png" /> if and only if:
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Later B.V. Gnedenko and W. Doeblin (1940) gave a description of the domain of attraction of a stable law with exponent $\alpha$, $0<\alpha<2$: $F(x)$ belongs to the domain of attraction of a non-degenerate stable law $V(x)$ with exponent $\alpha$ if and only if:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392018.png" /></td> </tr></table>
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$$\frac{F(-x)}{[1-F(x)+F(-x)]}\to\frac{c_1}{c_1+c_2}\quad\text{as }x\to\infty,$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392019.png" />, determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392020.png" />, and
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for some $c_1,c_2\geq0,c_1+c_2>0$, determined by $V(x)$, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392021.png" /></td> </tr></table>
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$$\frac{[1-F(x)+F(-x)]}{[1-F(tx)+F(-tx)]}\to t^\alpha\quad\text{as }x\to\infty,$$
  
for each constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392022.png" />. Restriction on the behaviour of the normalizing coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392024.png" /> leads to narrower classes of distribution functions for which the convergence in distribution (*) holds. The set of distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392025.png" /> for which (*) converges weakly, for a suitable choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392029.png" /> to a stable distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392030.png" /> with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392031.png" />, is called the normal domain of attraction for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392032.png" />. The normal domain of attraction of a normal distribution coincides with the set of non-degenerate distributions with a finite variance.
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for each constant $t>0$. Restriction on the behaviour of the normalizing coefficients $B_n$, $n=1,2,\dots,$ leads to narrower classes of distribution functions for which the convergence in distribution \eqref{*} holds. The set of distribution functions $F(x)$ for which \eqref{*} converges weakly, for a suitable choice of $A_n$, $c>0$ and $B_n=cn^{-1/2}$, $n=1,2,\dots,$ to a stable distribution function $V(x)$ with exponent $\alpha$, is called the normal domain of attraction for $V(x)$. The normal domain of attraction of a normal distribution coincides with the set of non-degenerate distributions with a finite variance.
  
The normal domain of attraction of a non-degenerate stable distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392033.png" /> with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392035.png" />) is formed by the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392036.png" /> for which
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The normal domain of attraction of a non-degenerate stable distribution function $V(x)$ with exponent $\alpha$ ($0<\alpha<2$) is formed by the functions $F(x)$ for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392037.png" /></td> </tr></table>
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$$\lim_{x\to-\infty}\frac{F(x)}{|x|^\alpha}=c_1\geq0,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392038.png" /></td> </tr></table>
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$$\lim_{x\to\infty}\frac{1-F(x)}{x^\alpha}=c_2\geq0$$
  
exist and are finite, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392039.png" /> are determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392040.png" />.
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exist and are finite, where $c_1,c_2$ are determined by $V(x)$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Gnedenko,   A.N. Kolmogorov,   "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Ibragimov,   Yu.V. Linnik,   "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"V.V. Petrov,   "Sums of independent random variables" , Springer (1975) (Translated from Russian)</TD></TR></table>
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{|
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|valign="top"|{{Ref|GK}}|| B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) {{MR|0062975}} {{ZBL|0056.36001}}
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|-
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|valign="top"|{{Ref|IL}}|| I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) {{MR|0322926}} {{ZBL|0219.60027}}
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|-
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|valign="top"|{{Ref|P}}|| V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}}
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|}

Latest revision as of 20:56, 1 January 2019

domain of attraction of a stable distribution

2020 Mathematics Subject Classification: Primary: 60F05 Secondary: 60E07 [MSN][ZBL]

The totality of all distribution functions $F(x)$ such that for a sequence of independent identically-distributed random variables $X_1,X_2,\dots,$ with distribution function $F(x)$ and for a suitable choice of constants $A_n$ and $B_n>0$, $n=1,2,\dots,$ the distribution of the random variable

$$\frac{\sum_{k=1}^nX_k-A_n}{B_n},\qquad n=1,2,\dots,\label{*}\tag{*}$$

converges weakly, as $n\to\infty$, to a non-degenerate distribution function $V(x)$, which is necessarily stable.

One of the fundamental problems in the theory of stable laws is the description of domains of attraction of stable laws. Thus, for the normal distribution, A.Ya. Khinchin, W. Feller and P. Lévy established in 1935 that $F(x)$ belongs to the domain of attraction of a normal law if and only if, as $x\to\infty$,

$$x^2\frac{\int\limits_{|y|>x}dF(y)}{\int\limits_{|y|<x}y^2\,dF(y)}\to0.$$

Later B.V. Gnedenko and W. Doeblin (1940) gave a description of the domain of attraction of a stable law with exponent $\alpha$, $0<\alpha<2$: $F(x)$ belongs to the domain of attraction of a non-degenerate stable law $V(x)$ with exponent $\alpha$ if and only if:

$$\frac{F(-x)}{[1-F(x)+F(-x)]}\to\frac{c_1}{c_1+c_2}\quad\text{as }x\to\infty,$$

for some $c_1,c_2\geq0,c_1+c_2>0$, determined by $V(x)$, and

$$\frac{[1-F(x)+F(-x)]}{[1-F(tx)+F(-tx)]}\to t^\alpha\quad\text{as }x\to\infty,$$

for each constant $t>0$. Restriction on the behaviour of the normalizing coefficients $B_n$, $n=1,2,\dots,$ leads to narrower classes of distribution functions for which the convergence in distribution \eqref{*} holds. The set of distribution functions $F(x)$ for which \eqref{*} converges weakly, for a suitable choice of $A_n$, $c>0$ and $B_n=cn^{-1/2}$, $n=1,2,\dots,$ to a stable distribution function $V(x)$ with exponent $\alpha$, is called the normal domain of attraction for $V(x)$. The normal domain of attraction of a normal distribution coincides with the set of non-degenerate distributions with a finite variance.

The normal domain of attraction of a non-degenerate stable distribution function $V(x)$ with exponent $\alpha$ ($0<\alpha<2$) is formed by the functions $F(x)$ for which

$$\lim_{x\to-\infty}\frac{F(x)}{|x|^\alpha}=c_1\geq0,$$

$$\lim_{x\to\infty}\frac{1-F(x)}{x^\alpha}=c_2\geq0$$

exist and are finite, where $c_1,c_2$ are determined by $V(x)$.

References

[GK] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[IL] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) MR0322926 Zbl 0219.60027
[P] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042
How to Cite This Entry:
Attraction domain of a stable distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attraction_domain_of_a_stable_distribution&oldid=20093
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article