Tauberian theorems and large deviations

References

• Balkema , A.A. , Geluk , J.L. and de Haan , L. 1979 . An extension of Karamata's Tauberian theorem and its connection with complementary convex functions . Quart. J. Math. , 30 : 385 – 416 .
   Google Scholar
• Biggins , J.D. and Bingham , N.H. 1993 . Large deviations in the supercritical branching process . Adv. Appl. Probab. , 25 : 759 – 772 .
   Google Scholar
• Bingham , N.H. 1971 . Limit theorems for occupation times for Markov processes . Z. Wahrschein , 17 : 1 – 22 .
   Web of Science ®Google Scholar
• Bingham , N.H. 2007 . Regular variation and probability: the early years . J. Comput. Appl. Math. , 200 : 357 – 363 .
   Google Scholar
• Bingham , N.H. , Goldie , C.M. and Teugels , J.L. 1989 . Regular Variation , 2nd ed. , Cambridge : Cambridge University Press . (1st ed. 1987)
   Google Scholar
• Bingham , N.H. and Teugels , J.L. 1975 . Duality for regularly varying functions . Quart. J. Math. , 26 ( 3 ) : 333 – 353 .
   Google Scholar
• de Bruijn , N.G. 1958 . Asymptotic Methods in Analysis North-Holland, Amsterdam
   Google Scholar
• Cartwright , M.L. 1956 . Integral Functions , Cambridge Tracts Math. vol. 44 , Cambridge : Cambridge University Press .
   Google Scholar
• Davies , P.L. 1976 . Tail behaviour for positive random variables with entire characteristic functions and completely regular growth . Z. Ang. Math. Mech. , 56 : T334 – T336 .
   Google Scholar
• Dupuis , P. and Ellis , R.S. 1997 . A Weak Convergence Approach to The Theory of Large Deviations , New York : Wiley .
   Google Scholar
• Ellis , R.S. 1985 . Entropy, Large Deviations and Statistical Mechanics , Grundl. Math. Wiss. vol. 271 , Berlin : Springer .
   Google Scholar
• Feller , W. 1971 . An Introduction to Probability Theory and Its Applications , 2nd ed. , vol. 2 , New York : Wiley .
   Google Scholar
• Fenton , P.C. 2005 . A glance at Wiman–Valiron theory . Contemp. Math. , 382 : 131 – 140 .
   Google Scholar
• Geluk , J.L. 1984 . On the relation between the tail probability and the moments of a random variable . Indagationes Math. A , 87 : 4091 – 405 .
   Google Scholar
• Geluk , J.L. 1986 . On a theorem in entire function theory and its applications in probability theory . J. Math. Anal. Appl. , 118 : 165 – 172 .
   Google Scholar
• Geluk , J.L. , de Haan , L. and Stadtmüller , U. 1986 . A Tauberian theorem of exponential type . Canad. J. Math. , 38 : 697 – 718 .
   Google Scholar
• Georgii , H.-O. 1988 . Gibbs Measures and Phase Transitions , Walter de Gruyter .
   Google Scholar
• Hayman , W.K. 1974 . The local growth of power series: a survey of the Wiman–Valiron method . Canad. Math. Bull. , 17 : 317 – 358 .
   Web of Science ®Google Scholar
• Kasahara , Y. 1978 . Tauberian theorems of exponential type . J. Math. Kyoto Univ. , 18 : 209 – 219 .
   Google Scholar
• Kasahara , Y. and Kosugi , N. 2002 . Large deviations around the origin for sums of non-negative i.i.d. random variables . Natur. Sci. Rep. Ochanomizu Univ. , 51 : 27 – 31 .
   Google Scholar
• Kasahara , Y. and Kosugi , N. 2002 . Remarks on Tauberian theorem of exponential type and Fenchel-Legendre transform . Osaka J. Math. , 39 : 613 – 620 .
   Google Scholar
• Kolmogorov , A.N. 1986 . Preface to The Law of Large Numbers Nauka, Moscow (in Russian)
   Google Scholar
• König , W. and Mörters , P. 2002 . Brownian intersection local times: upper tail asymptotics and thick points . Ann. Probab. , 30 : 1605 – 1656 .
   Google Scholar
• Korevaar , J. 2004 . Tauberian Theory: A Century of Developments , Grundl. Math. Wiss. vol. 329 , Berlin : Springer .
   Google Scholar
• Kosugi , N. 1999 . Tauberian theorem of exponential type and its application to multiple convolution . J. Math. Kyoto Univ. , 39 : 331 – 346 .
   Google Scholar
• Kosugi , N. 1999 . Tauberian theorem of exponential type on limits of oscillation . J. Math. Kyoto Univ. , 39 : 783 – 792 . Corr., 40 (2000), 303
   Google Scholar
• H. Leschke, P. Müller, and S. Warzel, A survey of rigorous results on random Schrödinger operators for amorphous solids, arXiv:cond-mat/0210708 v3 11 Dec 2003 (to appear, Markov Proc. Rel. Fields)
   Google Scholar
• Levin , B.Ya. Distribution of Zeros of Entire Functions Am. Math. Soc., Providence, RI, 1964 (Translations of Math. Monographs 5)
   Google Scholar
• Lukacs , E. 1970 . Characteristic Functions , 2nd ed. , London : Griffin . (1st ed. 1960)
   Google Scholar
• Lukacs , E. 1983 . Developments in Characteristic Function Theory , London : Griffin .
   Google Scholar
• Mikosch , T. 2004 . Non-Life Insurance Mathematics: An Introduction with Stochastic Processes , Universitext, Springer .
   Google Scholar
• Minlos , R.A. and Povsner , A.Ja. 1968 . Thermodynamic limits for entropy . Trudy Mosk. Mat. Obshch. , 17 : 243 – 272 .
   Google Scholar
• Nagai , H. 1977 . A remark on the Minlos–Povsner Tauberian theorem . Osaka Math. J. , 19 : 117 – 123 .
   Google Scholar
• Paley , R.E.A.C. and Wiener , N. 1934 . Fourier Transforms in the Complex Domain , Amer. Math. Soc. vol. XIX , AMS Colloquium Publications .
   Google Scholar
• Pitt , H.R. 1958 . Tauberian Theorems , Oxford : Oxford University Press .
   Google Scholar
• Rockafeller , R.T. 1970 . Convex Analysis , Princeton : Princeton University Press .
   Google Scholar
• Shiryaev , A.N. 1989 . Kolmogorov: life and creative activities . Ann. Probab. , 17 : 866 – 944 .
   Google Scholar
• Valiron , G. 1949 . Lectures on the General Theory of Integral Functions , New York : Dover .
   Google Scholar
• Wagner , E. 1966 . Taubersche Satz fiir die Mellin-Transformation . Wiss. Z. Martin-Luther Univ. Halle-Wittenberg Math.-Natur. Reihe , 15 : 617 – 624 .
   Google Scholar
• Wightman , A.S. 1979 . “ Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics. Introduction (ix–lxxxv), in R.B. Israel ” . In Convexity in the Theory of Lattice Gases , Princeton, NJ : Princeton University Press .
   Google Scholar
• Zolotarev , V.M. 1957 . Mellin–Stieltjes transform in probability theory . Th. Probab. Appl. , 2 : 433 – 460 .
   Google Scholar