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Lyapunov's theorem in probability theory is a theorem that establishes very general sufficient conditions for the convergence of the distributions of sums of independent random variables to the [[Normal distribution|normal distribution]]. The precise statement of Lyapunov's theorem is as follows: Suppose that the independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612001.png" /> have finite means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612002.png" />, variances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612003.png" /> and absolute moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612005.png" />, and suppose also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612006.png" /> is the variance of the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612007.png" />. Then if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l0612008.png" />,
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<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/l/l061/l061200/l0612009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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 +
Lyapunov's theorem in probability theory is a theorem that establishes very general sufficient conditions for the convergence of the distributions of sums of independent random variables to the [[Normal distribution|normal distribution]]. The precise statement of Lyapunov's theorem is as follows: Suppose that the independent random variables  $  X _ {1} , X _ {2} \dots $
 +
have finite means  $  {\mathsf E} X _ {k} $,
 +
variances  $  {\mathsf D} X _ {k} $
 +
and absolute moments  $  {\mathsf E} | X _ {k} - {\mathsf E} X _ {k} | ^ {2 + \delta } $,
 +
$  \delta > 0 $,
 +
and suppose also that  $  B _ {n} = \sum _ {k= 1}  ^ {n} {\mathsf D} X _ {k} $
 +
is the variance of the sum of  $  X _ {1} \dots X _ {n} $.  
 +
Then if for some  $  \delta > 0 $,
 +
 
 +
$$ \tag{1 }
 +
\lim\limits _ {n \rightarrow \infty } \
 +
 
 +
\frac{\sum _ { k= 1} ^ { n }
 +
{\mathsf E} | X _ {k} - {\mathsf E} X _ {k} | ^ {2 + \delta } }{B _ {n} ^ {1 + \delta / 2 } }
 +
  =  0 ,
 +
$$
  
 
the probability of the inequality
 
the probability of the inequality
  
<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/l/l061/l061200/l06120010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
x _ {1}  < \
 +
 
 +
\frac{\sum _ { k= 1} ^ { n }
 +
( X _ {k} - {\mathsf E} X _ {k} ) }{\sqrt {B _ {n} } }
 +
 
 +
< x _ {2}  $$
  
 
tends to the limit
 
tends to the limit
  
<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/l/l061/l061200/l06120011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
 
 +
\frac{1}{\sqrt {2 \pi } }
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120012.png" />, uniformly with respect to all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120014.png" />. Condition (1) is called the Lyapunov condition. Lyapunov's theorem was stated and proved by A.M. Lyapunov in 1901 and was the final step in research of P.L. Chebyshev, A.A. Markov and Lyapunov on conditions for the applicability of the [[Central limit theorem|central limit theorem]] of probability theory. Later, conditions were established that extend Lyapunov's conditions and that are not only sufficient but also necessary. A final solution of the question in this direction was obtained by S.N. Bernstein [S.N. Bernshtein], J.W. Lindeberg and W. Feller. The power of the method of characteristic functions was demonstrated for the first time in Lyapunov's theorem.
+
\int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
e ^ {- x  ^ {2} / 2 }  d x
 +
$$
  
Lyapunov also gave an upper bound (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120015.png" />) for the absolute value of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120016.png" /> between the probability of (2) and its approximate value (3). This bound can be expressed in the following form: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120017.png" />,
+
as  $  n \rightarrow \infty $,
 +
uniformly with respect to all values of  $  x _ {1} $
 +
and  $  x _ {2} $.
 +
Condition (1) is called the Lyapunov condition. Lyapunov's theorem was stated and proved by A.M. Lyapunov in 1901 and was the final step in research of P.L. Chebyshev, A.A. Markov and Lyapunov on conditions for the applicability of the [[Central limit theorem|central limit theorem]] of probability theory. Later, conditions were established that extend Lyapunov's conditions and that are not only sufficient but also necessary. A final solution of the question in this direction was obtained by S.N. Bernstein [S.N. Bernshtein], J.W. Lindeberg and W. Feller. The power of the method of characteristic functions was demonstrated for the first time in Lyapunov's theorem.
  
<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/l/l061/l061200/l06120018.png" /></td> </tr></table>
+
Lyapunov also gave an upper bound (for  $  \delta \leq  1 $)
 +
for the absolute value of the difference  $  \Delta $
 +
between the probability of (2) and its approximate value (3). This bound can be expressed in the following form: For  $  \delta < 1 $,
  
and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120019.png" />,
+
$$
 +
| \Delta |  \leq  C _ {1} L _ {n , \delta }  ,
 +
$$
  
<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/l/l061/l061200/l06120020.png" /></td> </tr></table>
+
and for  $  \delta = 1 $,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120022.png" /> are absolute constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120023.png" /> is the fraction (the Lyapunov fraction) under the limit sign in (1). See also [[Berry–Esseen inequality|Berry–Esseen inequality]].
+
$$
 +
| \Delta |  \leq  \
 +
C _ {2} L _ {n , 1 }  \
 +
\left |
 +
\mathop{\rm log} 
 +
\frac{1}{L _ {n,1} }
  
====References====
+
\right | ,
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Collected works" , '''1''' , Moscow-Leningrad  (1954)  pp. 157–176  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,  "Probability theory" , Moscow-Leningrad  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Feller,  "An introduction to probability theory and its applications" , '''2''' , Wiley  (1966)</TD></TR></table>
+
$$
  
 +
where  $  C _ {1} $
 +
and  $  C _ {2} $
 +
are absolute constants and  $  L _ {n , \delta }  $
 +
is the fraction (the Lyapunov fraction) under the limit sign in (1). See also [[Berry–Esseen inequality|Berry–Esseen inequality]].
  
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.M. Lyapunov, "Collected works", '''1''', Moscow-Leningrad (1954) pp. 157–176 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.N. Bernshtein, "Probability theory", Moscow-Leningrad (1946) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1966)</TD></TR></table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
Line 36: Line 88:
 
Lyapunov's theorems in potential theory are theorems on the behaviour of potentials and the solution of the [[Dirichlet problem|Dirichlet problem]], obtained by A.M. Lyapunov in 1886–1902 (see ).
 
Lyapunov's theorems in potential theory are theorems on the behaviour of potentials and the solution of the [[Dirichlet problem|Dirichlet problem]], obtained by A.M. Lyapunov in 1886–1902 (see ).
  
The theorem on the body of greatest potential: If there is a homogeneous body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120024.png" /> in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120025.png" />, the energy of whose [[Newton potential|Newton potential]], that is, the integral
+
The theorem on the body of greatest potential: If there is a homogeneous body $  T $
 +
in the Euclidean space $  \mathbf R  ^ {3} $,  
 +
the energy of whose [[Newton potential|Newton potential]], that is, the integral
  
<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/l/l061/l061200/l06120026.png" /></td> </tr></table>
+
$$
 +
E ( T)  = \int\limits _ { T } \int\limits _ { T }
 +
\frac{d x  d y }{| x - y | }
 +
,
 +
$$
  
 
attains its greatest value for a given volume, then this body is a ball.
 
attains its greatest value for a given volume, then this body is a ball.
Line 44: Line 102:
 
The integral
 
The integral
  
is the energy of a homogeneous mass distribution of density 1 on the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120027.png" />. Later, T. Carleman (1919) proved that a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120028.png" /> for which the energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120029.png" /> attains its greatest value for a given volume actually exists.
+
is the energy of a homogeneous mass distribution of density 1 on the body $  T $.  
 +
Later, T. Carleman (1919) proved that a body $  T $
 +
for which the energy $  E ( T) $
 +
attains its greatest value for a given volume actually exists.
 +
 
 +
First theorem on the normal derivatives of a double-layer potential: Let  $  S $
 +
be a closed Lyapunov surface in  $  \mathbf R  ^ {3} $,
 +
let  $  f ( y) $
 +
be the density of the mass distributed on  $  S $,
 +
and suppose that one of the following two conditions is satisfied: a)  $  f( y) $
 +
is continuous on  $  S $,
 +
and the exponent  $  \lambda = 1 $
 +
in the Lyapunov condition on the angle  $  \theta $
 +
between the normals to  $  S $
 +
at two points  $  y _ {1} , y _ {2} \in S $,
 +
that is,  $  | \theta | < A  | y _ {1} - y _ {2} | $ (see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]); or b)  $  f ( y) $
 +
is Hölder continuous with exponent 1, that is,  $  | f ( y _ {1} ) - f ( y _ {2} ) | < A  | y _ {1} - y _ {2} | $;
 +
then if for the [[Double-layer potential|double-layer potential]]
 +
 
 +
$$
 +
W ( x)  =  \int\limits _ { S } f ( y)
  
First theorem on the normal derivatives of a double-layer potential: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120030.png" /> be a closed Lyapunov surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120031.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120032.png" /> be the density of the mass distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120033.png" />, and suppose that one of the following two conditions is satisfied: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120034.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120035.png" />, and the exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120036.png" /> in the Lyapunov condition on the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120037.png" /> between the normals to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120038.png" /> at two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120039.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120040.png" /> (see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]); or b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120041.png" /> is Hölder continuous with exponent 1, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120042.png" />; then if for the [[Double-layer potential|double-layer potential]]
+
\frac{\cos ( y - x , n _ {y} ) }{| x - y | ^ {2} }
 +
\
 +
d y
 +
$$
  
<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/l/l061/l061200/l06120043.png" /></td> </tr></table>
+
one of the normal derivatives  $  d W _ {i} / d n _ {y _ {0}  } $
 +
inside  $  S $
 +
or  $  d W _ {e} / d n _ {y _ {0}  } $
 +
outside  $  S $
 +
at a point  $  y _ {0} \in S $
 +
exists, then the other also exists, and these derivatives coincide.
  
one of the normal derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120044.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120045.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120046.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120047.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120048.png" /> exists, then the other also exists, and these derivatives coincide.
+
Second theorem on the normal derivatives of a double-layer potential: Under the assumptions of the previous theorem, suppose also that the density  $  f ( y) $
 +
satisfies the Lyapunov condition
  
Second theorem on the normal derivatives of a double-layer potential: Under the assumptions of the previous theorem, suppose also that the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120049.png" /> satisfies the Lyapunov condition
+
$$
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
| f ( \rho , \phi ) - f ( y _ {0} ) | \
 +
d \phi  <  a \rho ^ {1 + \nu } ,\ \
 +
a , v > 0 ,
 +
$$
  
<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/l/l061/l061200/l06120050.png" /></td> </tr></table>
+
where  $  ( \rho , \phi , z ) $
 +
are cylindrical coordinates inside the Lyapunov sphere (see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]) with origin at a point  $  y _ {0} \in S $
 +
and  $  z $-axis directed along the normal  $  n _ {y _ {0}  } $.  
 +
Then the double-layer potential
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120051.png" /> are cylindrical coordinates inside the Lyapunov sphere (see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]) with origin at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120053.png" />-axis directed along the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120054.png" />. Then the double-layer potential
+
has both normal derivatives at $  y _ {0} $.
  
has both normal derivatives at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120055.png" />.
+
Theorem on the first derivatives of a simple-layer potential: Let  $  S $
 +
be a closed Lyapunov surface and suppose that the density  $  f ( y) $
 +
is Hölder continuous, that is,
  
Theorem on the first derivatives of a simple-layer potential: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120056.png" /> be a closed Lyapunov surface and suppose that the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120057.png" /> is Hölder continuous, that is,
+
$$
 +
| f ( y _ {1} ) - f ( y _ {2} ) |  < \
 +
A  | y _ {1} - y _ {2} |  ^  \lambda  ,\  0 < \lambda < 1 .
 +
$$
  
<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/l/l061/l061200/l06120058.png" /></td> </tr></table>
+
Then the first-order partial derivatives  $  \partial  V / \partial  x _ {i} $,
 +
$  i = 1 , 2 , 3 $,
 +
$  x = ( x _ {1} , x _ {2} , x _ {3} ) $,
 +
of the [[Simple-layer potential|simple-layer potential]]
  
Then the first-order partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120061.png" />, of the [[Simple-layer potential|simple-layer potential]]
+
$$
 +
V ( x)  = \int\limits _ { S } f ( y)
 +
\frac{d y }{| x - y | }
  
<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/l/l061/l061200/l06120062.png" /></td> </tr></table>
+
$$
  
are Hölder continuous with the same exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120063.png" /> in the closed interior domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120064.png" /> and closed exterior domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061200/l06120065.png" />.
+
are Hölder continuous with the same exponent $  \lambda $
 +
in the closed interior domain $  \overline{D}_ {i} $
 +
and closed exterior domain $  \overline{D}_ {e} $.
  
 
In this theorem the Hölder continuity was only stated by Lyapunov; the proof was completed by N.M. Gunther (see [[#References|[2]]]).
 
In this theorem the Hölder continuity was only stated by Lyapunov; the proof was completed by N.M. Gunther (see [[#References|[2]]]).
Line 75: Line 182:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A.M. Lyapunov,  , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 26–32  (In Russian)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A.M. Lyapunov,  , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 33–44  (In Russian)</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  A.M. Lyapunov,  "On certain questions connected with the Dirichlet problem" , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 45–47; 48–100  (In Russian)</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  A.M. Lyapunov,  , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 101–122  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Gunther,  "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
+
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A.M. Lyapunov,  , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 26–32  (In Russian)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A.M. Lyapunov,  , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 33–44  (In Russian)</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  A.M. Lyapunov,  "On certain questions connected with the Dirichlet problem" , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 45–47; 48–100  (In Russian)</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  A.M. Lyapunov,  , ''Collected works'' , '''1''' , Moscow  (1954)  pp. 101–122  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Gunther,  "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR>
 +
<TD valign="top">[a1]</TD> <TD valign="top">  J. Král,  "Integral operators in potential theory" , ''Lect. notes in math.'' , '''823''' , Springer  (1980)</TD></TR>
 +
</table>
  
 
''E.D. Solomentsev''
 
''E.D. Solomentsev''
 
====Comments====
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Král,  "Integral operators in potential theory" , ''Lect. notes in math.'' , '''823''' , Springer  (1980)</TD></TR></table>
 

Latest revision as of 14:20, 14 August 2023


Lyapunov's theorem in probability theory is a theorem that establishes very general sufficient conditions for the convergence of the distributions of sums of independent random variables to the normal distribution. The precise statement of Lyapunov's theorem is as follows: Suppose that the independent random variables $ X _ {1} , X _ {2} \dots $ have finite means $ {\mathsf E} X _ {k} $, variances $ {\mathsf D} X _ {k} $ and absolute moments $ {\mathsf E} | X _ {k} - {\mathsf E} X _ {k} | ^ {2 + \delta } $, $ \delta > 0 $, and suppose also that $ B _ {n} = \sum _ {k= 1} ^ {n} {\mathsf D} X _ {k} $ is the variance of the sum of $ X _ {1} \dots X _ {n} $. Then if for some $ \delta > 0 $,

$$ \tag{1 } \lim\limits _ {n \rightarrow \infty } \ \frac{\sum _ { k= 1} ^ { n } {\mathsf E} | X _ {k} - {\mathsf E} X _ {k} | ^ {2 + \delta } }{B _ {n} ^ {1 + \delta / 2 } } = 0 , $$

the probability of the inequality

$$ \tag{2 } x _ {1} < \ \frac{\sum _ { k= 1} ^ { n } ( X _ {k} - {\mathsf E} X _ {k} ) }{\sqrt {B _ {n} } } < x _ {2} $$

tends to the limit

$$ \tag{3 } \frac{1}{\sqrt {2 \pi } } \int\limits _ { x _ {1} } ^ { {x _ 2 } } e ^ {- x ^ {2} / 2 } d x $$

as $ n \rightarrow \infty $, uniformly with respect to all values of $ x _ {1} $ and $ x _ {2} $. Condition (1) is called the Lyapunov condition. Lyapunov's theorem was stated and proved by A.M. Lyapunov in 1901 and was the final step in research of P.L. Chebyshev, A.A. Markov and Lyapunov on conditions for the applicability of the central limit theorem of probability theory. Later, conditions were established that extend Lyapunov's conditions and that are not only sufficient but also necessary. A final solution of the question in this direction was obtained by S.N. Bernstein [S.N. Bernshtein], J.W. Lindeberg and W. Feller. The power of the method of characteristic functions was demonstrated for the first time in Lyapunov's theorem.

Lyapunov also gave an upper bound (for $ \delta \leq 1 $) for the absolute value of the difference $ \Delta $ between the probability of (2) and its approximate value (3). This bound can be expressed in the following form: For $ \delta < 1 $,

$$ | \Delta | \leq C _ {1} L _ {n , \delta } , $$

and for $ \delta = 1 $,

$$ | \Delta | \leq \ C _ {2} L _ {n , 1 } \ \left | \mathop{\rm log} \frac{1}{L _ {n,1} } \right | , $$

where $ C _ {1} $ and $ C _ {2} $ are absolute constants and $ L _ {n , \delta } $ is the fraction (the Lyapunov fraction) under the limit sign in (1). See also Berry–Esseen inequality.

References

[1] A.M. Lyapunov, "Collected works", 1, Moscow-Leningrad (1954) pp. 157–176 (In Russian)
[2] S.N. Bernshtein, "Probability theory", Moscow-Leningrad (1946) (In Russian)
[3] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1966)

Comments

References

[a1] R.G. Laha, V.K. Rohatgi, "Probability theory" , Wiley (1979)

Lyapunov's theorems in potential theory are theorems on the behaviour of potentials and the solution of the Dirichlet problem, obtained by A.M. Lyapunov in 1886–1902 (see ).

The theorem on the body of greatest potential: If there is a homogeneous body $ T $ in the Euclidean space $ \mathbf R ^ {3} $, the energy of whose Newton potential, that is, the integral

$$ E ( T) = \int\limits _ { T } \int\limits _ { T } \frac{d x d y }{| x - y | } , $$

attains its greatest value for a given volume, then this body is a ball.

The integral

is the energy of a homogeneous mass distribution of density 1 on the body $ T $. Later, T. Carleman (1919) proved that a body $ T $ for which the energy $ E ( T) $ attains its greatest value for a given volume actually exists.

First theorem on the normal derivatives of a double-layer potential: Let $ S $ be a closed Lyapunov surface in $ \mathbf R ^ {3} $, let $ f ( y) $ be the density of the mass distributed on $ S $, and suppose that one of the following two conditions is satisfied: a) $ f( y) $ is continuous on $ S $, and the exponent $ \lambda = 1 $ in the Lyapunov condition on the angle $ \theta $ between the normals to $ S $ at two points $ y _ {1} , y _ {2} \in S $, that is, $ | \theta | < A | y _ {1} - y _ {2} | $ (see Lyapunov surfaces and curves); or b) $ f ( y) $ is Hölder continuous with exponent 1, that is, $ | f ( y _ {1} ) - f ( y _ {2} ) | < A | y _ {1} - y _ {2} | $; then if for the double-layer potential

$$ W ( x) = \int\limits _ { S } f ( y) \frac{\cos ( y - x , n _ {y} ) }{| x - y | ^ {2} } \ d y $$

one of the normal derivatives $ d W _ {i} / d n _ {y _ {0} } $ inside $ S $ or $ d W _ {e} / d n _ {y _ {0} } $ outside $ S $ at a point $ y _ {0} \in S $ exists, then the other also exists, and these derivatives coincide.

Second theorem on the normal derivatives of a double-layer potential: Under the assumptions of the previous theorem, suppose also that the density $ f ( y) $ satisfies the Lyapunov condition

$$ \int\limits _ { 0 } ^ { {2 } \pi } | f ( \rho , \phi ) - f ( y _ {0} ) | \ d \phi < a \rho ^ {1 + \nu } ,\ \ a , v > 0 , $$

where $ ( \rho , \phi , z ) $ are cylindrical coordinates inside the Lyapunov sphere (see Lyapunov surfaces and curves) with origin at a point $ y _ {0} \in S $ and $ z $-axis directed along the normal $ n _ {y _ {0} } $. Then the double-layer potential

has both normal derivatives at $ y _ {0} $.

Theorem on the first derivatives of a simple-layer potential: Let $ S $ be a closed Lyapunov surface and suppose that the density $ f ( y) $ is Hölder continuous, that is,

$$ | f ( y _ {1} ) - f ( y _ {2} ) | < \ A | y _ {1} - y _ {2} | ^ \lambda ,\ 0 < \lambda < 1 . $$

Then the first-order partial derivatives $ \partial V / \partial x _ {i} $, $ i = 1 , 2 , 3 $, $ x = ( x _ {1} , x _ {2} , x _ {3} ) $, of the simple-layer potential

$$ V ( x) = \int\limits _ { S } f ( y) \frac{d y }{| x - y | } $$

are Hölder continuous with the same exponent $ \lambda $ in the closed interior domain $ \overline{D}_ {i} $ and closed exterior domain $ \overline{D}_ {e} $.

In this theorem the Hölder continuity was only stated by Lyapunov; the proof was completed by N.M. Gunther (see [2]).

These theorems served Lyapunov as a basis for the construction of a strict theory of solvability of the Dirichlet problem by the method of integral equations. A monograph of Gunther was devoted to the development of the ideas of Lyapunov (see [2]); for generalizations to potentials of a more general form see [3].

References

[1a] A.M. Lyapunov, , Collected works , 1 , Moscow (1954) pp. 26–32 (In Russian)
[1b] A.M. Lyapunov, , Collected works , 1 , Moscow (1954) pp. 33–44 (In Russian)
[1c] A.M. Lyapunov, "On certain questions connected with the Dirichlet problem" , Collected works , 1 , Moscow (1954) pp. 45–47; 48–100 (In Russian)
[1d] A.M. Lyapunov, , Collected works , 1 , Moscow (1954) pp. 101–122 (In Russian)
[2] N.M. Gunther, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian)
[3] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[a1] J. Král, "Integral operators in potential theory" , Lect. notes in math. , 823 , Springer (1980)

E.D. Solomentsev

How to Cite This Entry:
Lyapunov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_theorem&oldid=18024
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article