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A limit theorem for randomly stopped independent increment processes on separable metrizable groups. (English) Zbl 1143.60006

In the paper under review is presented a general central limit theorem for random stopping in the scheme of infinitesimal triangular arrays on a separable metrizable group. This theorem generalizes earlier results for normalized sequences of independent random variables on both separable Banach spaces G. Siegel, [Math. Nachr., 139, 139–153 (1988; Zbl 0066.60020)] and simply connected nilpotent Lie groups W. Hazod [J. Math. Sci., New York 93, No. 4, 531–542 (1999; Zbl 0942.60012)]. Let us present the Banach space generalization.
Let \((\xi_k)\) be a sequence of independent identically distributed random variables in a separable Banach space \(\mathbb{F}\). Let \(\xi_n(t)=A_n\sum_{1\leq k\leq k_nt}\xi_k\) for some sequence \((A_n)\) of linear operators on \(\mathbb{F}\) and let \(T_n/k_n\to D\) in probability, where \(D>0\) has distribution \(\varrho\). Let \(\xi_n(1)\) converges in distribution (\(\mathrm{d}\)) to some distribution \(\mu\) on \(\mathbb{F}\). Then for all sequences \(t_n\to t>0\) we have \[ A_n\sum\nolimits_{k=1}^{[T_nt_n]}\overset{\text{d}}\longrightarrow \int\nolimits_1^{\infty}\mu^{rt}\mathrm{d}\varrho(r). \]

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60G51 Processes with independent increments; Lévy processes
60G18 Self-similar stochastic processes
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[1] Aldous, Math. Proc. Cambridge Philos. Soc. 83 pp 117– (1978)
[2] Anscombe, Proc. Cambridge Philos. Soc. 48 pp 600– (1952)
[3] Becker–Kern, J. Theor. Probab. 16 pp 573– (2003)
[4] Becker–Kern, J. Appl. Anal. 10 pp 83– (2004)
[5] Blum, Z. Wahrsch. Verw. Gebiete 1 pp 389– (1963)
[6] Real Analysis and Probability (Wadsworth, Pacific Grove, 1989). · Zbl 0686.60001
[7] Feinsilver, Trans. Amer. Math. Soc. 242 pp 73– (1978)
[8] and , Wahrscheinlichkeitstheorie (Springer, Berlin, 1977).
[9] and , The Theory of Stochastic Processes I (Springer, New York, 1974).
[10] On limit theorems for a random number of random variables, in: Proceedings of the 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics, Tbilisi 1982, Lecture Notes in Mathematics Vol. 1021 (Springer, Berlin, 1983), pp. 167–176.
[11] and , Random Summation: Limit Theorems and Applications (CRC Press, Boca Raton, 1996).
[12] On some convolution semi- and hemigroups appearing as limit distributions of normalized products of groupvalued random variables, in: Analysis on Infinite-dimensional Lie Groups, edited by H. Heyer and J. Marion, Proceedings of the International Conference on Analysis on Infinite-dimensional Lie Groups, Marseille 1997 (World Scientific, New Jersey, 1998), pp. 104–121.
[13] Hazod, J. Math. Sci. (New York) 93 pp 531– (1999)
[14] and , Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups. Structural Properties and Limit Theorems (Kluwer, Dordrecht, 2001). · Zbl 1002.60002
[15] Heyer, J. Theor. Probab. 10 pp 1003– (1997)
[16] and , Operator Limit Distributions in Probability Theory (Wiley, New York, 1993). · Zbl 0850.60003
[17] and , Limit Distributions for Sums of Independent Random Vectors (Wiley, New York, 2001).
[18] Mogyoródi, Publ. Math. Inst. Hungar. Acad. Sci., Ser. A 7 pp 409– (1962)
[19] Nobel, J. Theor. Probab. 4 pp 261– (1991)
[20] Pap, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 pp 43– (2004)
[21] Rényi, Acta Math. Acad. Sci. Hungar. 9 pp 215– (1958)
[22] Rényi, Acta Math. Acad. Sci. Hungar. 11 pp 97– (1960)
[23] Probability Theory (North-Holland, Amsterdam, 1970).
[24] Scheffler, Publ. Math. Debrecen 47 pp 377– (1995)
[25] Regularly varying norming operators for domains of attraction on simply connected nilpotent Lie groups, unpublished manuscript, University of Dortmund (1997).
[26] Stable randomized limit theorems on simply connected nilpotent Lie groups, unpublished manuscript, University of Dortmund (2003).
[27] Siebert, Ann. Inst. H. Poincaré Probab. Statist. 20 pp 147– (1984)
[28] Siebert, Math. Z. 191 pp 73– (1986)
[29] Semistable convolution semigroups and the topology of contraction groups, in: Probability Measures on Groups IX, edited by H. Heyer, Proceedings of the International Conference on Probability Measures on Groups, Oberwolfach 1988, Lecture Notes in Mathematics Vol. 1379 (Springer, Berlin, 1989), pp. 325–343.
[30] Siegel, Statistics 17 pp 121– (1986)
[31] Siegel, Statistics 18 pp 437– (1987)
[32] Siegel, Math. Nachr. 139 pp 139– (1988)
[33] Siegel, Probab. Math. Statist. 13 pp 33– (1992)
[34] Silvestrov, Theory Probab. Appl. 17 pp 669– (1972)
[35] Limit Theorems for Randomly Stopped Stochastic Processes (Springer, London, 2004). · Zbl 1057.60021
[36] Tortrat, Ann. Inst. H. Poincaré, Nouv. Sér., Sect. B 1 pp 217– (1965)
[37] Stochastic Process Limits: An Introduction to Stochastic-process Limits and their Application to Queues (Springer, New York, 2002). · Zbl 0993.60001
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