# Local limit theorems

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.

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 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 [7], [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 [3], Chapt. 7); a number of papers have been devoted to local limit theorems for the case of an arbitrary stable distribution (see [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 (see [5], ).

#### References

 [1] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) [2] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) [3] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) [4] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) [5] S.Kh. Sirazhdinov, "Limit theorems for homogeneous Markov chains" , Tashkent (1955) (In Russian) [6a] 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) [6b] 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) [7] A.Ya. Khinchin, "Mathematical foundations of statistical mechanics" , Dover, reprint (1949) (Translated from Russian) [8] A.Ya. Khinchin, "Mathematical foundations of quantum statistics" , Moscow-Leningrad (1951) (In Russian)