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''in probability theory''

Limit theorems for densities, that is, theorems that establish the convergence of the densities of a sequence of distributions to the density of the limit distribution (if the given densities exist), or a classical version of local limit theorems, namely local theorems for lattice distributions, the simplest of which is the local [[Laplace theorem|Laplace theorem]].

Let  $  X _ {1} , X _ {2} \dots $
be a sequence of independent random variables that have a common distribution function  $  F ( x) $
with mean  $  a $
and finite positive variance  $  \sigma  ^ {2} $. 
Let  $  F _ {n} ( x) $
be the distribution function of the normalized sum

$$ 
Z _ {n}  =  
\frac{1}{\sigma \sqrt n }

\sum _ { j= } 1 ^ { n } 
( X _ {j} - a )
$$

and let  $  \Phi ( x) $
be the normal  $  ( 0 , 1 ) $-
distribution function. The assumptions ensure that  $  F _ {n} ( x) \rightarrow \Phi ( x) $
as  $  n \rightarrow \infty $
for any  $  x $. 
It can be shown that this relation does not imply the convergence of the density  $  p _ {n} ( x) $
of the distribution of the random variable  $  Z _ {n} $
to the normal density

$$ 

\frac{1}{\sqrt {2 \pi } }

e ^ {- x  ^ {2} / 2 } ,
$$

even if the distribution  $  F $
has a density. If  $  Z _ {n} $, 
for some  $  n = n _ {0} $, 
has a bounded density  $  p _ {n _ {0}  } ( x) $, 
then

$$ \tag{* }
p _ {n} ( x)  \rightarrow \ 

\frac{1}{\sqrt {2 \pi } }

e ^ {- x  ^ {2} / 2 }
$$

uniformly with respect to  $  x $. 
The condition that  $  p _ {n _ {0}  } ( x) $
is bounded for some  $  n _ {0} $
is necessary for (*) to hold uniformly with respect to  $  x $.

Let  $  X _ {1} , X _ {2} \dots $
be a sequence of independent random variables that have the same non-degenerate distribution, and suppose that  $  X _ {1} $
takes values of the form  $  b + N h $, 
$  N = 0 , \pm  1 , \pm  2 \dots $
with probability 1, where  $  h > 0 $
and  $  b $
are constants (that is,  $  X _ {1} $
has a [[Lattice distribution|lattice distribution]] with step  $  h $).

Suppose that  $  X _ {1} $
has finite variance  $  \sigma  ^ {2} $, 
let  $  a = {\mathsf E} X _ {1} $
and let

$$ 
P _ {n} ( N)  =  {\mathsf P} \left \{
\sum _ { j= } 1 ^ { n }  X _ {j} = n b + N h \right \} .
$$

In order that

$$ 
\sup _ { N } \ 
\left |

\frac{\sigma \sqrt n }{h}
 P _ {n} ( N) -

\frac{1}{\sqrt {2 \pi } }

 \mathop{\rm exp} \left \{
- 
\frac{1}{2}
 \left ( 
\frac{n b + N h - n a }{\sigma \sqrt n }
 \right )  ^ {2}
\right \} \right |  \rightarrow  0
$$

as  $  n \rightarrow \infty $
it is necessary and sufficient that the step  $  h $
should be maximal. This theorem of B.V. Gnedenko is a generalization of the local Laplace theorem.

Local limit theorems for sums of independent non-identically distributed random variables serve as a basic mathematical tool in classical statistical mechanics and quantum statistics (see [[#References|[7]]], [[#References|[8]]]).

Local limit theorems have been intensively studied for sums of independent random variables and vectors, together with estimates of the rate of convergence in these theorems. The case of a limiting normal distribution has been most fully investigated (see [[#References|[3]]], Chapt. 7); a number of papers have been devoted to local limit theorems for the case of an arbitrary [[Stable distribution|stable distribution]] (see [[#References|[2]]]). Similar investigations have been carried out for sums of dependent random variables, in particular for sums of random variables that form a [[Markov chain|Markov chain]] (see [[#References|[5]]], ).

====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><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.Kh. Sirazhdinov,   "Limit theorems for homogeneous Markov chains" , Tashkent  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top">  V.A. Statulyavichus,   "Limit theorems and asymptotic expansions for non-stationary Markov chains"  ''Litovsk. Mat. Sb.'' , '''1'''  (1961)  pp. 231–314  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top">  V.A. Statulyavichus,   "Limit theorems for sums of random variables that are connected in a Markov chain I"  ''Litovsk. Mat. Sb.'' , '''9'''  (1969)  pp. 345–362  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.Ya. Khinchin,   "Mathematical foundations of statistical mechanics" , Dover, reprint  (1949)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.Ya. Khinchin,   "Mathematical foundations of quantum statistics" , Moscow-Leningrad  (1951)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.N. Bhattacharya,   R. Ranga Rao,   "Normal approximations and asymptotic expansions" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Paulauskas,   "Approximation theory in the central limit theorem. Exact results in Banach spaces" , Kluwer  (1989)  (Translated from Russian)</TD></TR></table>