Abstract
Let X_{1}, X_{2}, ldots , X_{n} be independent integral-valued random variables, and let S_{n}=sum_{j=1}^{n}X_{j}. One of the interesting probabilities is the probability at a particular point, i.e., the density of S_{n}. The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate O (frac{1}{n} ) of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound frac{C}{sqrt{n}}. In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for S_{n}. Our constants are less complicated than before, and thus easy to use.
Highlights
Let X1, X2, . . . , Xn be independent integral-valued random variables with means μj and variances σj2 for j = 1, 2, . . . , n
One of the interesting probabilities is the probability at a particular point, i.e., P(Sn = k), where k = 1, 2, . . . . There are two density functions, i.e., discretized normal and normal, to approximate this probability
The various bounds for |ψ(t)| play a key role in the investigation of the rate of convergence in the local limit theorems
Summary
We improve the convergence rate and constants of error bounds in local limit theorem for Sn. Our constants are less complicated than before, and easy to use. The local limit theorem describes how the probability mass function of a sum of independent discrete random variables approaches the normal density. For sums of independent random variables, we can prove the local limit theorem by using the In 2018, Zolotukhin, Nagaev, and Chebotarev [12] gave the convergence with a constant of error bound in the case that Sn is a binomial.
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