We investigate conditions for the weak convergence of the maximum of sums of independent random processes in the space L p and present several applications to the asymptotic analysis of certain ω 2-type statistics.
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References
A. A. Borovkov, Probability Processes in Queueing Theory [in Russian], Nauka, Moscow (1972).
A. A. Borovkov, “Kolmogorov and boundary-value problems of probability theory,” Usp. Mat. Nauk, 59, No. 1, 91–102 (2004).
H. Cramer, Collective Risk Theory, Esselte, Stockholm (1955).
A. V. Skorokhod and N. P. Slobodenyuk, Limit Theorems for Random Walks [in Russian], Naukova Dumka, Kiev (1970).
V. Paulauskas, “On the distribution of the maximum of successive sums of independent identically distributed random vectors,” Liet. Mat. Rinkinys, 13, No. 2, 133–138 (1973).
V. Paulauskas and S. Steishunas, “On the rate of convergence of the distribution of the maximum of successive sums of independent nonidentically distributed random vectors to a limit law,” Liet. Mat. Rinkinys, 13, No. 2, 139–147 (1973).
J. Kiefer, “Skorokhod embedding of multivariate RV’s, and the sample DF,” Z. Wahrscheinlichkeitstheor. und verw. Geb., 24, No. 1, 1–35 (1972).
J. Lamperti, Probability [Russian translation], Nauka, Moscow (1973).
I. K. Matsak, “Limit theorem for the maximum of Gaussian independent random variables in the space C,” Ukr. Mat. Zh., 47, No. 7, 1006–1008 (1995).
L. S. Grinblat, “A limit theorem for measurable random processes and its applications,” Proc. Amer. Math. Soc., 61, No. 2, 371–376 (1976).
A. A. Borovkov, “Convergence of measures and random processes,” Usp. Mat. Nauk, 31, No. 2, 3–68 (1976).
P. Billingsley, Convergence of Probability Measures [Russian translation], Nauka, Moscow (1977).
H. P. Rosenthal, “On the subspaces of L p (p > 2) spanned by sequences of independent random variables,” Isr. J. Math., 8, No. 3, 273–303 (1970).
I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Vol. 1, Nauka, Moscow (1971).
N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985).
A. N. Kolmogorov, Probability Theory and Mathematical Statistics [in Russian], Nauka, Moscow (1986).
N. V. Smirnov, Probability Theory and Mathematical Statistics [in Russian], Nauka, Moscow (1970).
P. Gaenssler and W. Stute, “Empirical processes: a survey of results for independent and identically distributed random variables,” Ann. Probab., 7, No. 2, 193–243 (1979).
M. Csörgő and P. Révész, Strong Approximations in Probability and Statistics, Akadémiai Kiadó, Budapest (1981).
E. V. Khmaladze, “Some applications of the theory of martingales to statistics,” Usp. Mat. Nauk, 37, No. 6, 194–212 (1982).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1664–1674, December, 2008.
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Matsak, I.K. On some limit theorems for the maximum of sums of independent random processes. Ukr Math J 60, 1955–1967 (2008). https://doi.org/10.1007/s11253-009-0183-3
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DOI: https://doi.org/10.1007/s11253-009-0183-3