Article PDF
Avoid common mistakes on your manuscript.
References
Borovkov, A.A.: On the first passage time for one class of processes with independent increments. Theor. Probability Appl. 10, 331–334 (1965)
Chatterji, S.D.: An L P-convergence theorem. Ann. Math. Statist. 40, 1068–1070 (1965)
Chow, Y.S.: On the L P-convergence for n −1/p·S n , 0<p<2. Ann. Math. Statist. 42, 393–394 (1971)
Doob, J.L.: Stochastic processes. New York: Wiley 1953
Gikhman, I.I., Skorohod, A.V.: Introduction to the theory of random processes. Philadelphia: Saunders (1969)
Gundy, R.F., Siegmund, D.: On a stopping rule and the central limit theorem. Ann. Math. Statist. 38, 1915–1917 (1967)
Gut, A.: On the moments and limit distributions of some first passage times. Ann. Probability 2, 277–308 (1974)
Gut, A.: On convergence in r-mean of some first passage times and randomly indexed partial sums. Ann. Probability 2, 321–323 (1974)
Hall, W.J.: On Wald's equation in continuous time. J. Appl. Probability 7, 59–68 (1970)
Hille, E., Phillips, R.S.: Functional analysis and semi-groups. Amer. Math. Soc. 1957
Loève, M.: Probability theory (3rd ed.). New York: Van Nostrand 1953
Pyke, R., Root, D.: On convergence in r-mean of normalized partial sums. Ann. Math. Statist. 39, 379–381 (1968)
Rényi, A.: On the asymptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar. 8, 193–199 (1957)
Rogozin, B.A.: On the distribution of the first jump. Theor. Probability Appl. 9, 450–465 (1964)
Siegmund, D.O.: Some one-sided stopping rules. Ann. Math. Statist. 38, 1641–1646 (1967)
Zolotarev, V.M.: The first-passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theor. Probability Appl. 9, 653–662 (1964)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gut, A. On a.s. and r-mean convergence of random processes with an application to first passage times. Z. Wahrscheinlichkeitstheorie verw Gebiete 31, 333–341 (1975). https://doi.org/10.1007/BF00532871
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00532871