RESEARCH ARTICLE

Tail behavior of supremum of a random walk when Cramér’s condition fails

  • Changjun YU 1,2 ,
  • Yuebao WANG , 1
Expand
  • 1. School of Mathematical Sciences, Soochow University, Suzhou 215006, China
  • 2. School of Sciences, Nantong University, Nantong 226019, China

Received date: 02 Jun 2012

Accepted date: 25 Mar 2013

Published date: 01 Apr 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We investigate tail behavior of the supremum of a random walk in the case that Cramér’s condition fails, namely, the intermediate case and the heavy-tailed case. When the integrated distribution of the increment of the random walk belongs to the intersection of exponential distribution class and O-subexponential distribution class, under some other suitable conditions, we obtain some asymptotic estimates for the tail probability of the supremum and prove that the distribution of the supremum also belongs to the same distribution class. The obtained results generalize some corresponding results of N. Veraverbeke. Finally, these results are applied to renewal risk model, and asymptotic estimates for the ruin probability are presented.

Cite this article

Changjun YU , Yuebao WANG . Tail behavior of supremum of a random walk when Cramér’s condition fails[J]. Frontiers of Mathematics in China, 2014 , 9(2) : 431 -453 . DOI: 10.1007/s11464-013-0302-1

1
AndersenE S. On the collective theory of risk in case of contagion between claims. In: Transactions XVth International Congress of Actuaries, New York, 1957, II: 219-229

2
AsmussenS, FossS, KorshunovD. Asymptotics for sums of random variables with local subexponential behaviour. J Theoret Probab, 2003, 16: 489-518

DOI

3
AsmussenS, KalashnikovV, KonstantinidesD, KlüppelbergC, TsitsiashviliG. A local limit theorem for random walk maxima with heavy tails. Statist Probab Lett, 2002, 56: 399-404

DOI

4
BertoinJ, DoneyR A. Some asymptotic results for transient random walks. Adv Appl Probab, 1996, 28: 207-226

DOI

5
BorovkovA A. Stochastic Processes in Queueing. New York: Springer, 1976

DOI

6
ChenG, WangY, ChengF. The uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. Stoch Models, 2009, 25: 508-521

DOI

7
ChenY, WangL, WangY. Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J Math Anal Appl, 2013, 401: 114-129

DOI

8
ChengD, NiF, PakesA G, WangY. Some properties of the exponential distribution class with applications to risk theory. J Korean Statist Soc, 2012, 41: 515-527

DOI

9
ChistyakovV P. A theorem on sums of independent positive random variables and its applications to branching process. Theory Probab Appl, 1964, 9: 640-648

DOI

10
ChoverJ, NeyP, WaingerS. Functions of probability measures. J Anal Math, 1973, 26: 255-302

DOI

11
ChoverJ, NeyP, WaingerS. Degeneracy properties of subcritical branching processes. Ann Probab, 1973, 1: 663-673

DOI

12
CuiZ, WangY, WangK. Asymptotics for the moments of the overshoot and undershoot of a random walk. Adv Appl Probab, 2009, 41: 469-494

DOI

13
DenisovD, FossS, KorshunovD. On lower limits and equivalences for distribution tails of randomly stopped sums. Bernoulli, 2008, 14(2): 391-404

DOI

14
EmbrechtsP, GoldieC M. On closure and factorization properties of subexponential tails. J Aust Math Soc Ser A, 1980, 29: 243-256

DOI

15
EmbrechtsP, GoldieC M. On convolution tails. Stochastic Process Appl, 1982, 13: 263-278

DOI

16
EmbrechtsP, VeraverbekeN. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math Econom, 1982, 1: 55-72

DOI

17
FellerW. An Introduction to Probability Theory and Its Applications, Vol II. 2nd ed. New York: Wiley, 1971

18
FossS, KorshunovD, ZachryS. An Introduction to Heavy-tailed and Subexponential Distributions. New York: Springer, 2013

DOI

19
KlüppelbergC. Subexponential distributions and characterization of related classes. Probab Theory Related Fields, 1989, 82: 259-269

DOI

20
KlüppelbergC. Asymptotic ordering of distribution functions and convolution semigroups. Semigroup Forum, 1990, 40: 77-92

DOI

21
KlüppelbergC, StadtmüllerU. Ruin probabilities in the presence of heavy tails and interest rates. Scand Act J, 1998, (1): 49-58

22
KlüppelbergC, VillasenorJ A. The full solution of the convolution closure problem for convolution-equivalent distributions. J Math Anal Appl, 1991, 160: 79-92

DOI

23
KorshunovD. On distribution tail of the maximum of a random walk. Stochastic Process Appl, 1997, 72: 97-103

DOI

24
LeipusR, ŠiaulysJ. Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes. Insurance Math Econom, 2007, 40: 498-508

DOI

25
LeslieJ R. On the non-closure under convolution of the subexponential family. J Appl Probab, 1989, 26: 58-66

DOI

26
LinJ, WangY. Some new example and properties of O-subexponential distributions. Statist Probab Lett, 2012, 82: 427-432

DOI

27
PakesA G. Convolution equivalence and infinite divisibility. J Appl Probab, 2004, 41: 407-424

DOI

28
PitmanE J G. Subexponential distribution functions. J Aust Math Soc Ser A, 1980, 29: 337-347

DOI

29
ShimuraT, WatanabeT. Infinite divisibility and generalized subexponentiality. Bernoulli, 2005, 11: 445-469

DOI

30
SuC, ChenJ, HuZ. Some discussions on the class L(γ).J Math Sci, 2004, 122: 3416-3425

31
TangQ. Small Probability Problems in the Random Risk Model in Insurance and Financial. Ph D Thesis, University of Science and Technology of China, Hefei, 2001 (in Chinese)

32
TangQ. Asymptotics for the finite time ruin probability in the renewal model with consistent variation. Stoch Models, 2004, 20: 281-297

DOI

33
VeraverbekeN. Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stochastic Process Appl, 1977, 5: 27-37

DOI

34
WangK, WangY, GaoQ. Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol Comput Appl Probab, 2013, 15: 109-124

DOI

35
WangY, WangK. Asymptotics of the density of the supremum of a random walk with heavy-tailed increments. J Appl Probab, 2006, 43: 874-879

DOI

36
WangY, WangK. Equivalent conditions of asymptotics for the density of the supremum of a random walk in the intermediate case. J Theoret Probab, 2009, 22: 281-293

DOI

37
WangY, CuiZ, WangK, MaX. Uniform asymptotics of the finite-time ruin probability for all times. J Math Anal Appl, 2012, 390: 208-223

DOI

38
WangY, WangK. Random walks with non-convolution equivalent increments and their applications. J Math Anal Appl, 2011, 374: 88-105

DOI

39
WangY, YangY, WangK, ChengD. Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications. Insurance Math Econom, 2007, 42: 256-266

DOI

40
WatanabeT. Convolution equivalence and distributions of random sums. Probab Theory Related Fields, 2008, 142: 367-397

DOI

41
WatanabeT, YamamuroK. Ratio of the tail of an infinitely divisible distribution on the line to that of its Lévy measure. Electron J Probab, 2010, 15: 44-74

DOI

42
YangY, WangY. Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying tailed claims. Statist Probab Lett, 2010, 80: 143-154

DOI

43
YinC, ZhaoX, HuF. Ladder height and supremum of a random walk with applications in risk theory. Acta Math Sci Ser A Chin Ed, 2009, 29(1): 38-47 (in Chinese)

44
YuC, WangY, CuiZ. Lower limits and upper limits for tails of random sums supported on ℝ. Statist Probab Lett, 2010, 80: 1111-1120

45
ZongG. Finite-time ruin probability of a nonstandard compound renewal risk model with constant force of interest. Front Math China, 2010, 5(4): 801-809

DOI

Outlines

/