Large Deviations of Heavy-Tailed Sums with Applications in Insurance

First we give a short review of large deviation results for sums of i.i.d. random variables. The main emphasis is on heavy-tailed distributions. We stress more the methodology than the detailed calculations. Large deviation techniques are then applied to randomly indexed sums and shot noise processes. We also indicate the close relationship between large deviation results and the modeling of large insurance claims.

[1]  Claudia Klüppelberg,et al.  Explosive Poisson shot noise processes with applications to risk reserves , 1995 .

[2]  Richard A. Davis,et al.  Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance , 1995 .

[3]  S. V. Nagaev,et al.  Large Deviations of Sums of Independent Random Variables , 1979 .

[4]  Hans U. Gerber,et al.  An introduction to mathematical risk theory , 1982 .

[5]  Claudia Klüppelberg,et al.  Delay in claim settlement and ruin probability approximations , 1995 .

[6]  Anders Martin-Löf.  Entropy, a useful concept in risk theory , 1986 .

[7]  E. Omey,et al.  Second order behaviour of the tail of a subordinated probability distribution , 1986 .

[8]  Claudia Klüppelberg,et al.  Large deviations results for subexponential tails, with applications to insurance risk , 1996 .

[9]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[10]  Jean C. Walrand,et al.  Review of 'Large Deviation Techniques in Decision, Simulation, and Estimation' (Bucklew, J.A.; 1990) , 1991, IEEE Trans. Inf. Theory.

[11]  J. Grandell.  Aspects of Risk Theory , 1991 .

[12]  L. Haan,et al.  Residual Life Time at Great Age , 1974 .

[13]  Vladimir Vinogradov.  Refined Large Deviation Limit Theorems , 1994 .

[14]  Claudia Klüppelberg,et al.  LARGE DEVIATIONS OF HEAVY-TAILED RANDOM SUMS WITH APPLICATIONS IN INSURANCE AND FINANCE , 1997 .

[15]  MODERATE- AND LARGE-DEVIATION PROBABILITIES IN ACTUARIAL RISK THEORY , 1989 .

[16]  William S. Jewell,et al.  Gerber Hans U.: An Introduction to Mathematical Risk Theory Huebner Foundation Monograph No. 8. Homewood, Ill.: Richard D. Irwin Inc., 1980, xv + 164, paperbound, $ 15.95 , 1980, ASTIN Bulletin.

[17]  A contribution to the theory of large deviations for sums of independent random variables , 1967 .

[18]  P. Embrechts,et al.  Estimates for the probability of ruin with special emphasis on the possibility of large claims , 1982 .

[19]  L. Rozovskii.  Probabilities of large deviations on the whole axis , 1994 .

[20]  Boualem Djehiche.  A large deviation estimate for ruin probabilities , 1993 .

[21]  L. V. Osipov.  On Large Deviation Probabilities for Sums of Independent Random Vectors , 1979 .

[22]  James A. Bucklew,et al.  Large Deviation Techniques in Decision, Simulation, and Estimation , 1990 .

[23]  A. V. Nagaev.  On a Property of Sums of Independent Random Variables , 1978 .

[24]  V. V. Petrov.  Sums of Independent Random Variables , 1975 .

[25]  Y. Linnik.  Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. III , 1961 .

[26]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[27]  V. V. Petrov.  On the Probabilities of Large Deviations for Sums of Independent Random Variables , 1965 .

[28]  S. Resnick.  Adventures in stochastic processes , 1992 .

[29]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[30]  C. Heyde.  On Large Deviation Problems for Sums of Random Variables which are not Attracted to the Normal Law , 1967 .

[31]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

[32]  S. Nagaev.  Some Limit Theorems for Large Deviations , 1965 .

[33]  PAUL EMBRECHTS,et al.  Modelling of extremal events in insurance and finance , 1994, Math. Methods Oper. Res..

[34]  Harald Cram'er,et al.  Sur un nouveau théorème-limite de la théorie des probabilités , 2018 .

[35]  U. Stadtmüller,et al.  Generalized regular variation of second order , 1996, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.