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 be a sequence of independent random variables that have a common distribution function with mean and finite positive variance . Let be the distribution function of the normalized sum

and let be the normal -distribution function. The assumptions ensure that as for any . It can be shown that this relation does not imply the convergence of the density of the distribution of the random variable to the normal density

even if the distribution has a density. If , for some , has a bounded density , then

(*)

uniformly with respect to . The condition that is bounded for some is necessary for (*) to hold uniformly with respect to .

Let be a sequence of independent random variables that have the same non-degenerate distribution, and suppose that takes values of the form , with probability 1, where and are constants (that is, has a lattice distribution with step ).

Suppose that has finite variance , let and let

In order that

as it is necessary and sufficient that the step 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)