Harmonic renewal measures
Tóm tắt
If C is a distribution function on (0, ∞) then the harmonic renewal function associated with C is the function
$$G(x) = \sum\limits_1^\infty {n^{ - 1} } C^{(n)} (x)$$
. We link the asymptotic behaviour of G to that of 1−C. Applications to the ladder index and the ladder height of a random walk are included.
Tài liệu tham khảo
Bojanic, R., Seneta, E.: A unified theory of regularly varying sequences. Math. Z. 134, 91–106 (1973)
Borovkov, A.A.: Stochastic Processes in Queueing Theory. Berlin-Heidelberg-New York: Springer 1976
De Haan, L.: On regular variation and its applications to the weak convergence of sample extremes. Math. Centre Tracts. Amsterdam 1970
De Haan, L.: An Abel-Tauber theorem for Laplace transforms. J. London Math. Soc. 13, 537–542 (1976)
Doney, R.A.: A note on a condition satisfied by certain random walks. J. Appl. Probab. 14, 843–849 (1977)
Embrechts, P.: A second order theorem for Laplace transforms, J. London Math. Soc. 17, 102–106 (1978)
Emery, D.J.: On a condition satisfied by certain random walks. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 125–139 (1975)
Feller, W.: An introduction to probability theory and its applications, Volume II, 2nd ed. New York: Wiley 1971
Feller, W.: Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119 (1949)
Geluk, J.L., De Haan, L.: On functions with small differences. Preliminary report (1980)
Heyde, C.C.: Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37, 699–710 (1966)
Lai, T.L.: Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab. 4, 51–66 (1976)
Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88, 380–387 (1958)
Mohan, N.R.: Teugels' renewal theorem and stable laws. Ann. Probab. 4, 863–868 (1976)
Nagaev, S.V.: Some renewal theorems. Theory Prob., 13, 547–563 (1968)
Omey, E.: A note on discrete renewal sequences. Forthcoming.
Prabhu, N.V.: Stochastic Processes. New York: MacMillan 1965
Rogosin, B.A.: On the distribution of the first jump. Theory Prob., 9, 450–465 (1964)
Rogosin, B.A.: The distribution of the first ladder moment and height and fluctuations of a random walk. Theory Prob., 16, 575–595 (1971)
Rosén, B.: On the asymptotic distribution of sums of random variables. Ark. Math., 4, 323–332 (1960)
Seneta, E.: Regularly varying functions. Lecture Notes Math. 508, Berlin-Heidelberg-New York: Springer 1974
Sinai, Ya.G.: On the distribution of the first positive sum for a sequence of independent random variables. Theory Prob., 2, 122–129 (1957)
Smith, W.L.: On the weak law of large numbers and the generalized elementary renewal theorem. Pacific J. Math., 22, 171–188 (1967)
Stam, A.J.: Regular variation of the tail of a subordinated probability distribution. Adv. in Appl. Prob., 5, 308–327 (1973)
Spitzer, F.: Principles of random walk. New York: Van Nostrand 1964
Weissman, I.: A note on Bojanic-Seneta Theory of regularly varying sequences. Math. Z., 151, 29–30 (1976)