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)

References

[a1]	R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions" , Wiley (1976)
[a2]	V. Paulauskas, "Approximation theory in the central limit theorem. Exact results in Banach spaces" , Kluwer (1989) (Translated from Russian)