Summary
The ladder timeN and ladder heightH of a random walk {S n ,n≧1} as a pair (N, H) lie in the domain of attraction of a bivariate stable law ifS 1 is in a domain of attraction, as was shown by Greenwood et al. (1982). In this paper we prove a converse. IfP(S n >0) converges and (N, H) lies in a bivariate domain of attraction thenS 1 is also in a domain of attraction.
References
Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab.7, 705–766 (1975)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Cambridge: Cambridge University Press, 1989
Darling, D.A.: The maximum of sums of random variables. Trans. Am. Math. Soc.83, 164–169 (1956)
Doney, R.A.: A note on a condition satisfied by certain random walks. J. Appl. Probab.14, 843–849 (1977)
Doney, R.A.: Spitzer's condition for asymptotically symmetric random walk. J. Appl. Probab.17, 856–859 (1980)
Doney, R.A.: On Wiener-Hopf factorization and the distribution of extrema for certain stable processes. Ann. Probab.15, 1352–1362 (1987)
Doney, R.A.: A bivariate local limit theorem. J. Multivariate Anal.36, 95–102 (1991)
Embrechts, P., Hawkes, J.: A limit theorem for the tails of discrete infinitely divisble laws with applications to fluctuation theory. J. Aust. Math. Soc., Ser. A32, 412–422 (1982)
Emery, D.J.: On a condition satisfied by certain random walks. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 125–139 (1975)
Feller, W.: An introduction to probability theory and its applications, vol. II, 1st edn. New York: Wiley 1966
Greenwood, P.E., Omey, E., Teugels, J.L.: Harmonic renewal measures and bivariate domains of attraction in fluctuation theory. Z. Wahrscheinlichkeitstheor. Verw. Geb.61, 527–539 (1982)
Griffin, P.: Matrix normalised sums of independent identically distributed random vectors. Ann. Probab.14, 224–246 (1986)
Grübel, R.: Tail-behaviour of ladder height distributions in random walks. J. Appl. Probab.22, 705–709 (1985)
Hahn, M.G., Klass, M.J.: The generalized domain of attraction of spherically symmetric stable laws onR d. In: Weron, A. (ed.) Probability theory on vector spaces, II. (Lect. Notes Math., vol. 828, pp. 52–81) Berlin Heidelberg New York: Springer 1979
Hahn, M.G., Klass, M.J.: Affine normability of partial sums of i.i.d. random vectors. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 479–506 (1985)
Heyde, C.C.: On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Probab.6, 419–429 (1969)
Pitman, E.J.G.: On the behaviour of the characteristic function of a probability distribution in the neighborhood of the origin. J. Am. Math. Soc. Ser. A 8, 422–443 (1968)
Resnick, S.I., Greenwood, P.: A bivariate stable characterization and domains of attraction. J. Multivariate Anal.9, 206–221 (1979)
Rogozin, B.A.: On the distribution of the first jump. Theory Probab. App.9, 450–465 (1964)
Rogozin, B.A.: On the distribution of the first ladder moment and height and fluctuations of a random walk. Theory Probab. Appl.16, 575–595 (1971)
Rossberg, H.J.: Limit theorems for identically distributed summands assuming the convergence of the distribution functions on a half axis. Theory Probab. Appl.24, 693–711 (1979)
Rvaceva, E.L.: On the domains of attraction of multidimensional distributions. Sel. Trans. Math. Stat. Probab. Theory2, 183–205 (1962)
Sinai, Ya G.: On the distribution of the first positive sum for a sequence of independent random variables. Theory Probab. Appl.2, 122–129 (1957)
Spitzer, F.: Principles of random walk. Princeton N.J.: Van Nostrand 1964
Veraverbeke, N.: Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stochastic Processes Appl.5, 27–37 (1977)
Zolotarev, V.M.: Mellin-Stieltjes transform in probability theory. Theory Probab. Appl.2, 433–460 (1957)
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Doney, R.A., Greenwood, P.E. On the joint distribution of ladder variables of random walk. Probab. Th. Rel. Fields 94, 457–472 (1993). https://doi.org/10.1007/BF01192558
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DOI: https://doi.org/10.1007/BF01192558
Keywords
- Stochastic Process
- Random Walk
- Probability Theory
- Mathematical Biology
- Joint Distribution