121
Views
1
CrossRef citations to date
0
Altmetric
Research Articles

The local limit theorem for general weighted sums of Bernoulli random variables

, &
Pages 4918-4926 | Received 18 Jun 2022, Accepted 27 Mar 2023, Published online: 08 Apr 2023

References

  • Benedicks, M. 1975. An estimate of the modulus of the characteristic function of a lattice distribution with application to remainder term estimates in local limit theorems. Annals of Probability 3:162–5.
  • Doob, J. L. 1953. Stochastic processes. New York: Chapman and Hall.
  • Giuliano, R., and M. J. G. Weber. 2016. Local limit theorems in some random models from number theory. Stochastic Analysis and Applications 34:941–60. doi: 10.1080/07362994.2016.1191994.
  • Giuliano, R., and M. J. G. Weber. 2017. Approximate local limit theorems with effective rate and application to random walks in random scenery. Bernoulli 23 (4B):3268–310. doi: 10.3150/16-BEJ846.
  • Kammoo, P., K. Laipaporn, and K. Neammanee. 2023. Local limit theorems without assuming finite third moment. Journal of Inequalities and Applications 21:1–16. doi: 10.1186/s13660-023-02928-y.
  • Mcdnald, D. R. 2005. The local limit theorem: A historical perspective. Journal of the Iranian Statistical Society 4 (2):73–86.
  • Prokhorov, Y. V., and Y. A. Rozanov. 1973. Theory of probabilities. 2nd ed., 494. Moscow: Nauka (in Russian).
  • Siripraparat, T., and K. Neammanee. 2021a. An improvement of convergence rate in the local limit theorem for integral-valued random variables. Journal of Inequalities and Applications 57:1–18. doi: 10.1186/s13660-021-02590-2.
  • Siripraparat, T., and K. Neammanee. 2021b. A local limit theorem for Poisson binomial random variables. Scienceasia 47:111–6. doi: 10.2306/scienceasia1513-1874.2021.006.
  • Statulyavichus, V. A. 1965. Limit theorems for densities and asymptotic decompositions for distributions of sums of independent random variables. Theory of Probability and Its Applications 10:582–95. doi: 10.1137/1110074.
  • Ushakov, N. 1997. Lower an upper bounds for characteristic functions. Journal of Mathematical Sciences 84:1179–89. doi: 10.1007/BF02398431.
  • Ushakov, N. G. 1999. Selected topics in characteristic functions. 395. Utrecht, The Netherlands: VSP.
  • Vladimirovich, G. B. 1948. On a local limit theorem of the theory of probability. Uspekhi Matematicheskikh Nauk 3 (25):187–94.
  • Zhang, Z. 2007. An upper bound for characteristic functions of lattice distributions with applications to survival probabilities of quantum states. Journal of Physics A: Mathematical and Theoretical 40 (1):131–7. doi: 10.1088/1751-8113/40/1/007.
  • Zhang, Z. 2012. Bound for characteristic functions and laplace tranforms of probability distributions. Theory of Probability and Its Applications, 56:350–8. doi: 10.1137/S0040585X97985479.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.