Summary
The local limit problem is investigated for sequences (p n ) of probability densities with stable limit densitiesq α having characteristic exponent α∈(0, 2).
It is shown that certain continuity-properties (Hölder-continuity) are necessary and - under appropriate additional conditions-sufficient for\(\mathop {\sup }\limits_x |p_n (x) - q_\alpha (x)| \to 0\) asn→∞. In this sence the speed of convergence is also studied.
References
Basu, S.K., Maejima, M., Patra, N.K.: A non-uniform rate of convergence in local limit theorems concerning variables in the domain of normal attraction of a stable law. Yokohama Math. J.27, 63–72 (1979)
Basu, S.K., Maejima, M.: A local limit theorem for attractions under a stable law. Math. Proc. Camb. Philos. Soc.87, 179–187 (1980)
Christoph, G.: Weighted sums of random variables attracted to stable laws. Probab. Th. Rel. Fields71, 335–340 (1986)
Davydov, Yu.A., Shukri, F.: Local limit theorems for weighted sums of independent random variables (in russ.). Vestn. Leningr. Univ. Math.13, 140–142 (1975)
Gnedenko, B.V.: On the local limit theorem for independent and identically distributed summands (in russ.). Wiss. Zt. Humboldt-Univ. Berlin, Math.-Nat. Reihe,3, No.4, 287–293 (1953/54)
Heinrich, L.: Non-uniform estimates and asymptotic expansions of the remainder in the central limit theorem form-dependent random variables. Math. Nachr.115, 7–20 (1984)
Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Groningen, Wolters-Noordhoff 1971
Lukacs, E.: Characteristic functions, 2nd edition. London: Griffin, 1970
Macht, W., Wolf, W.: On the local central limit theorem. In: Limit theorems in probability theory and related fields. TU Dresden, Wissenschaft, Theorie und Praxis, 69–87, 1987
Maejima, M.: A non-uniform estimate in the local limit theorem for densities I, II. Yokohama Math. J.26, 83–89, 119–135 (1978)
Maejima, M.: The remainder term in the local limit theorem for independent random variables. Tokyo J. Math. 3, 311–329 (1980)
Nagaev, A.V.: Some remarks on multidimensional local limit theorems (in russ). Mat. zametki14, 559–563 (1973)
Paulauskas, V.: The estimation of the remainder term in limit theorems with a limiting stable distribution (in russ.). Litov. Mat. Sb.14, 165–187 (1974)
Petrov, V.V.: Sums of independent random variables. Berlin: Akademie-Verlag 1975
Pipiras, V.: A local limit theorem for densities (in russ.). Litov. Mat. Sb.14, 231–232 (1974)
Rossberg, H.-J., Jesiak, B., Siegel, G.: Analytic methods of probability theory. Berlin: Akademie-Verlag 1985
Saulis, L.: On large deviations for the probability density of sums of independent random variables. Probab. Theory Math. Stat.2, 541–559, VNU Science Press 1986
Shervashidse, T.L.: On multidimensional local limit theorems for densities (in russ.) In: Limit theorems and stochastic equations, 12–53, Tblissi 1984
Statulevičius, V.A.: Local limit theorems and asymptotic expansions for inhomogeneous Markov chains (in russ.). Litov. Mat. Sb.1, 231–313 (1961)
Wolf, W.: Lokale Grenzwertsätze für gewichtete Summen. Statistics16, 243–247 (1985)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Macht, W., Wolf, W. The local limit problem and Hölder-continuity. Probab. Th. Rel. Fields 82, 295–305 (1989). https://doi.org/10.1007/BF00354765
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00354765
Keywords
- Probability Density
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Stable Limit