Randomly stopped maximum and maximum of sums with consistently varying distributions
Volume 4, Issue 1 (2017), pp. 65–78
Pub. online: 6 March 2017
Type: Research Article
Open Access
Open Access
Received
10 December 2016
10 December 2016
Revised
23 January 2017
23 January 2017
Accepted
27 January 2017
27 January 2017
Published
6 March 2017
6 March 2017
Abstract
Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let $S_{0}:=0$ and $S_{n}:=\xi _{1}+\xi _{2}+\cdots +\xi _{n}$ for $n\geqslant 1$. We consider conditions for random variables $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution functions of the random maximum $\xi _{(\eta )}:=\max \{0,\xi _{1},\xi _{2},\dots ,\xi _{\eta }\}$ and of the random maximum of sums $S_{(\eta )}:=\max \{S_{0},S_{1},S_{2},\dots ,S_{\eta }\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.
References
Albin, J.M.P.: A note on the closure of convolution power mixtures (random sums) of exponential distributions. J. Aust. Math. Soc. 84, 1–7 (2008). MR2469263. doi:10.1017/S1446788708000104
Cai, J., Tang, Q.: On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Probab. 41, 117–130 (2004). MR2036276. doi:10.1017/S002190020001408X
Chen, Y., Ng, K.W., Yuen, K.C.: The maximum of randomly weighted sums with long tails in insurance and finance. Stoch. Anal. Appl. 29, 1033–1044 (2011). MR2847334. doi:10.1080/07362994.2011.610163
Chen, Y., Yuen, K.C.: Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25, 76–89 (2009). MR2494614. doi:10.1080/15326340802641006
Chistyakov, V.P.: A theorem on sums of independent positive random variables and its application to branching processes. Theory Probab. Appl. 9, 640–648 (1964). doi:10.1137/1109088
Cline, D.B.H.: Convolutions of distributions with exponential and subexponential tails. J. Aust. Math. Soc. A 43, 347–365 (1987). MR0904394. doi:10.1017/S1446788700029633
Danilenko, S., Šiaulys, J.: Randomly stopped sums of not identically distributed heavy tailed random variables. Stat. Probab. Lett. 113, 84–93 (2016). MR3480399. doi:10.1016/j.spl.2016.03.001
Danilenko, S., Paškauskaitė, S., Šiaulys, J.: Random convolution of inhomogeneous distributions with $\mathcal{O}$ exponential tail. Mod. Stoch., Theory Appl. 3, 79–94 (2016). doi:10.15559/16-VMSTA52
Denisov, D., Foss, S., Korshunov, D.: Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli 16, 971–994 (2010). MR2759165. doi:10.3150/10-BEJ251
Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Process. Appl. 13, 263–278 (1982). MR0671036. doi:10.1016/0304-4149(82)90013-8
Embrechts, P., Omey, E.: A property of longtailed distributions. J. Appl. Probab. 21, 80–87 (1984). MR0732673. doi:10.1017/S0021900200024396
Kaas, R., Tang, Q.: Note on the tail behavior of random walk maxima with heavy tails and negative drift. N. Am. Actuar. J. 7, 57–61 (2003). doi:10.1080/10920277.2003.10596103
Kizinevič, E., Sprindys, J., Šiaulys, J.: Randomly stopped sums with consistently varying distributions. Mod. Stoch., Theory Appl. 3, 165–179 (2016). doi:10.15559/16-VMSTA60
Klüppelberg, C.: Subexponential distributions and integrated tails. J. Appl. Probab. 25, 132–141 (1988). doi:10.1017/S0021900200040705
Leipus, R., Šiaulys, J.: Closure of some heavy-tailed distribution classes under random convolution. Lith. Math. J. 52, 249–258 (2012). doi:10.1007/s10986-012-9171-7
Ng, K.W., Tang, Q., Yang, H.: Maxima of sums of heavy-tailed random variables. ASTIN Bull. 32, 43–55 (2002). MR1928012. doi:10.2143/AST.32.1.1013
Wang, D., Tang, Q.: Maxima of sums and random sums for negatively associated random variables with heavy tails. Stat. Probab. Lett. 68, 287–295 (2004). MR2083897. doi:10.1016/j.spl.2004.03.011
Watanabe, T., Yamamuro, K.: Ratio of the tail of an infinitely divisible distribution on the line to that of its Lévy measure. Electron. J. Probab. 15, 44–75 (2010). doi:10.1214/EJP.v15-732
Xu, H., Foss, S., Wang, Y.: On closedness under convolution and convolution roots of the class of long-tailed distributions. Extremes 18, 605–628 (2015). MR3418770. doi:10.1007/s10687-015-0224-2
Zhang, J., Cheng, F., Wang, Y.: Tail behavior of random sums of negatively associated increments. J. Math. Anal. Appl. 376, 64–73 (2011). doi:10.1016/j.jmaa.2010.10.001
