Central limit theorems revisited. (English) Zbl 0963.60016
The authors prove four central limit theorems (CLT). The first CLT is for a triangular array of random elements taking values in a real separable Hilbert space, where the components of the array are row-wise independent and have finite second moments, the covariance operator of the row sum satisfies mild convergence requirements, and the array satisfies the Lindeberg condition along the evaluation at an orthonormal basis. The second CLT is for a real-valued martingale difference array for which the row sum of conditional variances converges in probability to a constant \(\sigma^{2}\) and a conditional Lindeberg condition is satisfied. The third CLT is an extension of the second one to multi-dimensions, and the fourth CLT is an extension of the third one to the Hilbert space setting.
The first CLT is an extension of Theorem 4.3.2 of V. Fabian and J. Hannan [“Introduction to probability and mathematical statistics” (1985; Zbl 0558.62001)]. The second CLT is Theorem VIII.1 of D. Pollard [“Convergence of stochastic processes” (1984; Zbl 0544.60045)] strengthened to include \(\sigma^{2}=0\). The technique, namely convergence in distribution via uniform approximation, is used to extend the results from finite to infinite dimensions. Extension to random elements taking values in a Banach space with a Schauder basis is indicated.
The first CLT is an extension of Theorem 4.3.2 of V. Fabian and J. Hannan [“Introduction to probability and mathematical statistics” (1985; Zbl 0558.62001)]. The second CLT is Theorem VIII.1 of D. Pollard [“Convergence of stochastic processes” (1984; Zbl 0544.60045)] strengthened to include \(\sigma^{2}=0\). The technique, namely convergence in distribution via uniform approximation, is used to extend the results from finite to infinite dimensions. Extension to random elements taking values in a Banach space with a Schauder basis is indicated.
Reviewer: Jun Kawabe (Wakasato)
MSC:
| 60F05 | Central limit and other weak theorems |
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
Keywords:
central limit theorem; Hilbert space; Lindeberg condition; martingale difference array; weak convergenceReferences:
| [1] | Araujo, A., Giné, E., 1980. The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.; Araujo, A., Giné, E., 1980. The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York. · Zbl 0457.60001 |
| [2] | Basu, A. K., Uniform and nonuniform estimates in the clt for Banach valued dependent random variables, J. Multivariate Anal., 25, 153-163 (1988) · Zbl 0681.60012 |
| [3] | Chung, K.L., 1970. A Course in Probability Theory. Academic Press, New York.; Chung, K.L., 1970. A Course in Probability Theory. Academic Press, New York. |
| [4] | Dunford, N., Schwartz, J.T., 1958. Linear Operators. Part I: General Theory. Wiley Classics Library Edition 1988. Wiley, New York.; Dunford, N., Schwartz, J.T., 1958. Linear Operators. Part I: General Theory. Wiley Classics Library Edition 1988. Wiley, New York. |
| [5] | Enflo, P., A counter-example to the approximation problem in Banach spaces, Acta Math., 103, 309-317 (1973) · Zbl 0267.46012 |
| [6] | Fabian, V., Hannan, J., 1985. Introduction to Probability and Mathematical Statistics. Wiley, New York.; Fabian, V., Hannan, J., 1985. Introduction to Probability and Mathematical Statistics. Wiley, New York. · Zbl 0558.62001 |
| [7] | Gaenssler, P., Haeusler, E., 1986. On martingale central limit theory. In: Taqqu, M., Eberlein, E. (Eds.), Dependence in Probability and Statistics; A Survey of Recent Results. Birkhauser, Boston.; Gaenssler, P., Haeusler, E., 1986. On martingale central limit theory. In: Taqqu, M., Eberlein, E. (Eds.), Dependence in Probability and Statistics; A Survey of Recent Results. Birkhauser, Boston. · Zbl 0632.60028 |
| [8] | Garling, D. J.H., Functional central limit theorems in Banach spaces, Ann. Probab., 4, 600-611 (1976) · Zbl 0343.60014 |
| [9] | Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and Its Applications. Academic Press, New York.; Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and Its Applications. Academic Press, New York. · Zbl 0462.60045 |
| [10] | Helland, I. S., Central limit theorems for martingales with discrete or continuous time, Scand. J. Statist., 9, 79-94 (1982) · Zbl 0486.60023 |
| [11] | Parthasarathy, K.R., 1967. Probability Measures on Metric Spaces. Academic Press, New York.; Parthasarathy, K.R., 1967. Probability Measures on Metric Spaces. Academic Press, New York. · Zbl 0153.19101 |
| [12] | Pollard, D., 1984. Convergence of Stochastic Processes. Springer, New York.; Pollard, D., 1984. Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045 |
| [13] | Xie, Y., Limit theorems of Hilbert valued semimartingales and Hilbert valued martingale measures, Stochastic Process. Appl., 59, 277-295 (1995) · Zbl 0836.60004 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
