Random sums of independent random vectors attracted by (semi)-stable hemigroups

• Autorzy: Becker-Kern P.
• Języki publikacji: EN

Abstrakty

EN

Let (Xn) be a sequence of independent not necessarily identically distributed random vectors belonging to the domain of attraction of a stable or semistable hemigroup, i.e. for an increasing sampling sequence (kn) such that kn+1/kn → c ≥ 1 and linear operators An, the normalized sums [wzór] converge in distribution uniformly on compact subsets of {0 ≤ s < t} to some full probability μs,t. Suppose that (Tn) is a sequence of positive integer valued random variables such that Tn/kn converges in probability to some positive random variable, where we do not assume (Xn) and (Tn) to be independent. Then weak limit theorems of random sums, where the sampling sequence (kn) is replaced by random sample sizes (Tn), are presented.

Słowa kluczowe

EN

random sum; semistable hemigroup; random sample size; Anscombe-condition; operator selfdecomposability; random centering

PL

przypadkowa suma; próbka wielkości losowej; Anscombe-warunek; losowe centrowanie

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2004
• Tom: Vol. 10, nr 1
• Strony: 83--104
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Becker-Kern P.

• Fachbereich Mathematik Universität Dortmund D-44221 Dortmund Germany,

Bibliografia

• [1] Anscombe, F. J., Large-sample theory of sequential estimation, Proc. Cambridge Philos. Soc. 48 (1952), 600-607.
• [2] Becker-Kern, P., Limit theorems with random sample size for generalized domains of semistable attraction, J. Math. Sci. Ill (2002), 3820-3829.
• [3] Becker-Kern, P., Stable and semistable hemigroups: Domains of attraction and self- decomposability, J. Theoret. Probab. 16 (2003), 573-598.
• [4] Blum, J. R., Hanson, D. L., Rosenblatt, J. I., On the central limit theorem for the sum of a random number of random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/1963), 389-393.
• [5] Breiman, L., Probability, Addison Wesley Publishing Company, Reading, Mass.- London-Don Mills, Ont., 1968.
• [6] Csörgő, M., Rychlik, Z., Weak convergence of sequences of random elements with random indices, Math. Proc. Cambridge Philos. Soc. 88 (1980), 171-174.
• [7] Gleser, L. J., On limiting distributions for sums of a random number of independent random vectors, Ann. Math. Statist. 40 (1969), 935-941.
• [8] Gnedenko, B. V., On limit theorems for a random number of random variables, Proceedings of the 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics (Tbilisi, 1982), Lecture Notes in Math. 1021 (1983), Springer, Berlin, 167-176.
• [9] Guiasu, S., On the asymptotic distribution of the sequences of random variables with random indices, Ann. Math. Statist. 42 (1971), 2018-2028.
• [10] Hazod, W., Some new limit theorems for vector space- and group-valued random variables, J. Math. Sci. 93 (1999), 531-542.
• [11] Hazod, W., Stable hemigroups and mixing of generating functionals, J. Math. Sci. Ill (2002), 3830-3840.
• [12] Hudson, W. N., Mason, J. D., Operator-self-similar processes in a finite dimensional space, Trans. Amer. Math. Soc. 273 (1982), 281-297.
• [13] Kimbleton, S. R., A simple proof of a random stable limit theorem, J. Appl. Probab. 7 (1970), 502-504.
• [14] Krajka, A., Rychlik, Z., Necessary and sufficient conditions for weak convergence of random sum of independent random variables, Comment. Math. Univ. Carolin. 34 (1993), 465-482.
• [15] Kubacki, K. S., Szynal, D., Weak convergence of randomly indexed sequences of random variables, Bull. Polish Acad. Sci. Math. 32 (1985), 201-210.
• [16] Maejima, M., Sato, K., Semi-selfsimilar processes, J. Theoret. Probab. 12 (1999), 347-373.
• [17] Maejima, M., Sato, K., Watanabe, T., Distributions of selfsimilar and semiselfsimilar processes with independent increments, Statist. Probab. Lett. 47 (2000), 395-401.
• [18] Meerschaert, M. M., Scheffler, H. P., Limit Distributions for Sums of Independent Random Vectors, John Wiley & Sons, Inc., New York, 2001.
• [19] Meerschaert, M. M., Scheffler, H. P., Spectral decomposition for generalized domains of semistable attraction J. Theoret. Probab. 10 (1997), 51-71.
• [20] Mogyoródi, J., A central limit theorem for the sum of a random number of independent random variables, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 409-424.
• [21] Rényi, A., On the central limit theorem for the sum of a random number of independent random variables, Acta Math. Acad. Sci. Hungar. 11 (1960), 97-102.
• [22] Rényi, A., Probability Theory, North-Holland Publishing Co., Amsterdam-London, 1970.
• [23] Sato, K., Self-similar processes with independent increments, Probab. Theory Related Fields 89 (1991), 285-300.
• [24] Sato, K., Yamamuro, K., On selfsimilar and semi-selfsimilar processes with independent increments, J. Korean Math. Soc. 35 (1998), 207-224.
• [25] Siegel, G., Convergence of stochastic processes with random parameters, Statistics 17 (1986), 121-138.
• [26] Silvestrov, D. S., Remarks on the limit of a composite random function, Theory Probab. Appl. 17 (1972), 669-677.
• [27] Wittenberg, H., Limiting distributions of random sums of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 7-18.