Asymptotic and non-asymptotic estimates for multivariate Laplace integrals

  • Maria Rosaria Formica Università Parthenope di Napoli
  • Eugeny Ostrovsky Bar-Ilan University, Department of Mathematic and Statistic
  • Leonid Sirota Bar-Ilan University, Department of Mathematic and Statistic
Keywords: Laplace or exponential integrals, Fenchel-Morau theorem, random variable and random vector, exponential and ordinary tail of distribution


We derive bilateral  asymptotic  as well as non-asymptotic estimates for the multivariate Laplace integrals. Furthermore, we provide multidimensional Tauberian theorems for exponential integrals.


D.R. Bagdasarov and E.I. Ostrovsky, Reversion of Chebyshev’s Inequality, Probab. Theory Appl., vol. 40, no. 4, 737–742, 1996.

C. Bennet and R. Sharpley, Interpolation of operators, Academic Press, Inc., Boston, MA, 1988.

N. H. Bingham, Tauberian theorems and large deviations, ArXiv:0712.3410v1 [math.PR] 20 Dec 2007.

M. Broniatowski and A. Fuchs, Tauberian Theorems, Chernoff Inequality and the Tail Behavior of Finite Convolution of Distribution Function, Adv. Math., vol. 116, no. 1, 12–33, 1995.

V.V. Buldygin and Yu.V. Kozachenko, Metric Characterization of Random Variables and Random Processes, Translations of Mathematics Monograph, AMS, vol.188, 1998.

H. Chen, Evaluation of the Laplace integral, Internat. J. Math.Ed. Sci. Tech., vol. 35, no. 5, 773–777, 2004.

H. Chernoff, A career in statistics, In X. Lin, C. Genest, D.L. Banks, G. Molenberghs, D.W. Scott, J-L. Wang, Past, Present, and Future of Statistical Science. CRC Press. p. 35. ISBN 9781482204964, 2014.

H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math.Statistics, vo. 23, 493–507, 1952.

P.L. Davies, Tail probabilities for positive random variables with entire characteristic functions of very regular growth, Z. Angew. Math. Mech., vol. 56, 334–336, 1976.

P. Eichelsbacher and L. Knichel, Fine asymptotics for models with Gamma type moments, ArXiv:1710.06484v1 [math.PR] 17 Oct 2017.

M.V. Fedoryuk, The saddle-point method, Moscow, Nauka (In Russian), 1977.

J.L. Geluk, L. de Haan and U. Stadtmüller, A Tauberian theorem of exponential type, Canad. J. Math. vol. 38, no. 3, 697-718, 1986.

J.L. Geluk, On the relation between the tail probability and the moments of a random variable, Nederl. Akad. Wetensch. Indag. Math., vol. 46, no. 4, 401–405, 1984.

S. Janson, Further examples with moments of Gamma type, arXiv:1204.5637v2, 6 Feb 2013.

S. Janson, Moments of Gamma type and the Brownian supremum process area, Probab. Surv., vol. 7, 1–52, 2010.

Y. Kasahara, Tauberian theorems of exponential type, J. Math. Kyoto Univ., vol. 18, no. 2, 209–219, 1978.

Y. Kasahara and N. Kosugi, Remarks on Tauberian theorem of exponential type and Fenchel-Legendre transform, Osaka J. Math., vol. 39, no. 3, 613–619, 2002.

V.N. Kolokoltsov, T.M. Lapinski, Multivariate Laplace approximation with estimated error and application to limit theorems,ArXiv:1502.03266v5 [math.PR] 17 Jul 2018.

J. Korevaar, Tauberian theory: a century of developments, Grundlehren der Mathematischen Wissenschaften, vol. 329, Springer-Verlag, Berlin, 2004.

N. Kosugi, Tauberian theorem of exponential type and its application to multiple convolution, J. Math. Kyoto Univ., vol. 39, no. 2,331–346, 1999.

Yu.V. Kozachenko, Yu.Yu. Mlavets and N.V. Yurchenko, Weak convergence of random processes from spaces FΨ(Ω), Stat. Optim. Inf. Comput. vol. 6, no. 2, 266–277, 2018.

Yu.V. Kozachenko and E.I. Ostrovsky, The Banach Spaces of random Variables of subgaussian Type, Theory of Probab. and Math.Stat., (in Russian). Kiev, KSU, vol. 32, 43–57, 1985.

Yu.V. Kozachenko, E.I. Ostrovsky and L. Sirota, Relations between exponential tails, moments and moment generating functions for random variables and vectors ArXiv:1701.01901v1 [math.FA] 8 Jan 2017.

J.L. López and P.J. Pagola, An explicit formula for the coefficients of the saddle point method, Constr. Approx. vol. 33,no. 2,145–162, 2011.

V.P. Maslov and M.V. Fedoryuk, Logarithmic Asymptotic behavior of the Laplace integrals, Mathematical Notes, vol. 30, no. 5,763–768, 1981.

D.M. Mason, An extended version of the Erdös-Rényi strong law of large numbers, Ann. Probab., vol. 17, no. 1, 252–265, 1989.

G. Nemes, An explicit formula for the coefficients in Laplace’s method, Constr. Approx., vol. 38, no. 3, 471–487, 2013.

G. Nemes, An Extension of Laplace’s Method, Constr. Approx.,, 2018.

E.I. Ostrovsky, Exponential estimations for Random Fields and its applications, (in Russian). Moscow-Obninsk, OINPE, 1999.

E. Ostrovsky and L. Sirota, Vector rearrangement invariant Banach spaces of random variables with exponential decreasing tails of distributions, ArXiv:1510.04182v1 [math.PR] 14 Oct 2015.

E. Ostrovsky and L. Sirota, Non-asymptotical sharp exponential estimates for maximum distribution of discontinuous random fields,ArXiv:1510.08945v1 [math.PR] 30 Oct 2015.

E.I. Ostrovsky, About supports of probability measures in separable Banach spaces, Soviet Math., Doklady (in Russian), vol. 255,no. 6, 836–838, 1980.

E. Ostrovsky and L. Sirota, Inversion of Tchebychev-Tchernov inequality, ArXiv:1711.06896v1 [math.PR] 18 Nov 2017.

E. Seneta, Regularly Varying Functions, Springer-Verlag, New York, 1976.

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971.

A. Tauber, Ein Satz aus der Theorie der unendlichen Reihen, Monatsh. Math. Phys., vol. 8, no. 1, 273–277, 1987.

A.L.Yakimiv, Probabilistic applications of Tauberian theorems, Modern probability and statistics,VSP,Leiden,ISBN:9067644374, 2005.

A. Zhang and Y. Zhou, A Non-asymptotic, Sharp, and User friendly Reverse Chernoff-Cramer Bound. ArXiv:1810.09006v1 [math.PR] 21 Oct 2018.

How to Cite
Formica, M. R., Ostrovsky, E., & Sirota, L. (2019). Asymptotic and non-asymptotic estimates for multivariate Laplace integrals. Statistics, Optimization & Information Computing, 7(4), 759-778.
Research Articles