LOCAL LIMIT THEOREM, ITS APPLICATIONS AND RESULTS

Аннотация

This article provides information about the local limit theorem, its definition, mathematical expression, and itsdifference from thecentral limittheorem. In addition, thearticle discusses someoftheresults derived from the theorem and itsareas ofapplication. Keywords : Events, experiments, sequence of experiments, probability of an event, local limit theorem ,central limittheorem The local limit theorem is one of the most important theorems in probability theory. This theorem helps to determine the fine details of the distribution of sums of random variables. The development of the local limit theorem occupies an important place in the history of probability theory. The initial forms of this theorem were developed by mathematicians in the late 19th and early 20th centuries. It was initially developed on the basis of the law of large numbers and the central limit theorem, and later began to be widely used in solving complex problems in the field of random processes and probability density. The definition and mathematical expression of the theorem are as follows : According to the classical definition of the local limit theorem: if X ,X ,...,X there are 1 2 n mutually independent random variables, which have the same distribution and are of finite variance, then the probability density function of the sum of these random variables approaches theGaussian (or normal)distribution densityfor increasing numbers. In mathematical language, if S = X + X +...+ X the sumis ,then: n 1 2 n 1 (k-nm)2 P(S =k)» exp - , (1.1) n 2pns2 2ns2 where and s2are mthemean and variance, respectively. It is important to understand the difference between the local limit theorem and the central limit theorem. According to the central limit theorem, the sum of a large number of independent random variables will converge to a normal distribution, but this convergence will occur at a point in the middle of the general probability distribution. In other words, the central limit theorem considers the convergence of the probability density function. The local limit theorem, on the other hand, deals with the probabilities themselves, that is, it allows us to determine the probability that the sum of random variables will take certain values. This allows us to study the 8 Volume 4, issue 4, 2025 finer details of the convergence process, since it does not examine the density of the distribution, buttheprobabilities corresponding tothevalues ofthevariable. Basic concepts and mathematical basis: To understand the local limit theorem, it is necessary to know some basic concepts, mathematical expressions, and important principles of probability theory. A random variable is an event whose value can change. For example, a coin toss may result in a "heads" or a "number". Mathematically, a random variable is described by X its possible values and the probabilities that correspond to them. The concept of a probability distribution is used to describe theseprobabilities. Probabilitydistributionscan bediscrete orcontinuous: Discrete distribution: A random variable that takes on discrete values, such as whole numbers. Each value has adefined probability, such as in acoin toss.P ( X = gerb ) = 0.5 Continuous distribution : A random variable takes on continuous values. In this case, the distribution isdefined usingaprobability density function. For example, anormal distribution. Basic concepts of mathematical probability theory: 1. Mathematical expectation (mean value): It is defined as the average value of a random variableand E(X)is determined by. E(X)= xP(X =x) (1.2) i i i or E(X)= xf(x)dx (1.3) - where is f(x)theprobabilitydensity function. 2. Dispersion: Provides information about how much a random variable deviates from the mean. Var(X)=E[(X -E(X))2] (1.4) 3. Standard deviation: It is thesquare root ofthevariance and measures thedegree ofspread. s= Var(X) (1.5) 9 Volume 4, issue 4, 2025 The main idea in the proof of the local limit theorem is to estimate the distribution of the sum of random variables by means of the characteristic function. This process requires the use of Fourier transforms and asymptotic methods, since it can be difficult to calculate the distribution exactlywhen thesum ofrandom variables is large. Another important challenge is to determine the degree of asymptotic convergence that occurs during the normalization of the sum and the removal of the mean values. This convergence is estimated based onthelargenumber ofvalues. Several basic mathematical methods are used in theproofof thelocal limittheorem:  Fourier analysis: Calculating the probability density using the characteristic function and Fourier inversion plays akey roleinproving thelocal limit theorem.  Asymptotic analysis: Asymptotic methods are used to evaluate the convergence of a distribution to a normal distribution. This method ensures the accuracy of the proof of the local limittheorem.  Markov chains and probability chain theory: This theory is used to evaluate the sta