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 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 , Chapt. 7); a number of papers have been devoted to local limit theorems for the case of an arbitrary stable distribution (see ). 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 , ).
|||B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)|
|||I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)|
|||V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)|
|||Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)|
|||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)|
|||A.Ya. Khinchin, "Mathematical foundations of statistical mechanics" , Dover, reprint (1949) (Translated from Russian)|
|||A.Ya. Khinchin, "Mathematical foundations of quantum statistics" , Moscow-Leningrad (1951) (In Russian)|
|[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)|
Local limit theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_limit_theorems&oldid=11793