zbMATH — the first resource for mathematics

Moment conditions in strong laws of large numbers for multiple sums and random measures. (English) Zbl 1386.60176
Summary: The validity of the strong law of large numbers for multiple sums $$S_{\mathbf{n}}$$ of independent identically distributed random variables $$Z_{\mathbf{k}}$$, $$\mathbf{k}\leq \mathbf{n}$$, with $$r$$-dimensional indices is equivalent to the integrability of $$|Z|(\log^+|Z|)^{r-1}$$, where $$Z$$ is the generic summand. We consider the strong law of large numbers for more general normalizations, without assuming that the summands $$Z_{\mathbf{k}}$$ are identically distributed, and prove a multiple sum generalization of the Brunk-Prohorov strong law of large numbers. In the case of identical finite moments of order $$2q$$ with integer $$q \geq 1$$, we show that the strong law of large numbers holds with the normalization $$(n_1 \cdots n_r)^{1/2}(\log n_1 \cdots \log n_r)^{1/(2q)+\varepsilon}$$ for any $$\varepsilon > 0$$.
The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.
MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F15 Strong limit theorems 60D05 Geometric probability and stochastic geometry
Full Text:
References:
 [1] Brunk, H. D., The strong law of large numbers, Duke Math. J., 15, 181-195, (1948) · Zbl 0030.20003 [2] Daley, D. J.; Vere-Jones, D., An Introduction To the Theory of Point Processes. Vol. I: Elementary Theory and Methods, (2003), Springer New York · Zbl 1026.60061 [3] Daley, D. J.; Vere-Jones, D., An Introduction To the Theory of Point Processes. Vol. II: General Theory and Structure, (2008), Springer New York · Zbl 0657.60069 [4] Dharmadhikari, S. W.; Fabian, V.; Jogdeo, K., Bounds on the moments of martingales, Ann. Math. Statist., 39, 1719-1723, (1968) · Zbl 0196.19202 [5] Fazekas, I.; Klesov, O., A general approach to the strong laws of large numbers, Teor. Veroyatn. Primen., 45, 568-583, (2000) · Zbl 0991.60021 [6] Gaposhkin, V. F., Multiparametric strong law of large numbers for homogeneous random fields, Uspekhi Mat. Nauk, 36, 6(222), 197-198, (1981) · Zbl 0479.60058 [7] Kallenberg, O., Foundations of Modern Probability, (2002), Springer-Verlag New York · Zbl 0996.60001 [8] Khoshnevisan, D., Multiparameter Processes, (2002), Springer-Verlag New York · Zbl 1005.60005 [9] Klesov, O. I., The strong law of large numbers for homogeneous random fields, Teor. Veroyatn. I Mat. Statist., 25, 29-40, (1981), 166 · Zbl 0462.60033 [10] Klesov, O. I., A new method for the strong law of large numbers for random fields, Theory Stoch. Process., 4, 1-2, 122-128, (1998) · Zbl 0941.60056 [11] Klesov, O., Limit Theorems for Multi-Indexed Sums of Random Variables, Vol. 71, (2014), Springer Heidelberg · Zbl 1318.60005 [12] Lagodowski, Z. A., Strong laws of large numbers for $$\mathbb{B}$$-valued random fields, Discrete Dyn. Nat. Soc., (2009), pages Art. ID 485412, 12 [13] Petrov, V. V., (Limit Theorems of Probability Theory, Oxford Studies in Probability, vol. 4, (1995), The Clarendon Press, Oxford University Press New York) · Zbl 0826.60001 [14] Prohorov, Yu. V., On the strong law of large numbers, Izv. Akad. Nauk SSSR. Ser. Mat., 14, 523-536, (1950) · Zbl 0040.07301 [15] Smythe, R. T., Strong laws of large numbers for $$r$$-dimensional arrays of random variables, Ann. Probab., 1, 164-170, (1973) · Zbl 0258.60026 [16] Smythe, R. T., Ergodic properties of marked point processes in $$R^r$$, Ann. Inst. H. Poincaré. Sect. B (N.S.), 11, 109-125, (1975) · Zbl 0307.60047 [17] Son, T. C.; Thang, D. H., The brunk-prokhorov strong law of large numbers for fields of martingale differences taking values in a Banach space, Statist. Probab. Lett., 83, 1901-1910, (2013) · Zbl 1281.60030 [18] Tempel’man, A. A., Ergodic theorems for general dynamical systems, Trudy Moskov. Mat. Obšč., 26, 94-132, (1972), (in Russian) · Zbl 0281.28008 [19] von Bahr, B.; Esseen, C.-G., Inequalities for $$r$$th absolute moment of a sum of random variables, $$1 \leq r \leq 2$$, Ann. Math. Statist., 36, 299-303, (1965) · Zbl 0134.36902 [20] Wichura, M. J., Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters, Ann. Math. Statist., 40, 681-687, (1969) · Zbl 0214.17701 [21] Zakai, M., Some classes of two-parameter martingales, Ann. Probab., 9, 255-265, (1981) · Zbl 0462.60055 [22] Zygmund, A., An individual ergodic theorem for non-commutative transformations, Acta Sci. Math. (Szeged), 14, 103-110, (1951) · Zbl 0045.06403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.