zbMATH — the first resource for mathematics

A note on the almost sure central limit theorem. (English) Zbl 0691.60016
For broad classes of sequences \(\{X_ j\), \(j\geq 1\}\) of weakly dependent r.v.’s almost sure invariance principles (ASIP) are known e.g. in the form \[ \sum_{j\leq n}X_ j-\sum_{j\leq n}Y_ j=o(n^{1/2})\text{ with probability 1 }(n\to \infty), \] where \(\{Y_ j\), \(j\geq 1\}\) is a sequence of i.i.d. N(0,1) r.v.’s. The behaviour of the k-th partial sums \(S_ k\) of \(\{X_ j\), \(j\geq 1\}\) is studied by the help of a summation method (logarithmic mean), if an ASIP is supposed. A new proof of an almost sure central limit theorem (without ergodic theory), and some extensions and further remarks on known special results in this field are given.
Reviewer: L.Paditz

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
40G99 Special methods of summability
Full Text: DOI
[1] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[2] Brosamler, G.A., An almost everywhere central limit theorem, Math. proc. Cambridge philos. soc., 104, 561-574, (1988) · Zbl 0668.60029
[3] Brosamler, G.A., Un théorème limite presque sur pour LES mesures d’occupation du mouvement brownien sur une variété riemannienne compacte, C.R. acad. sci. Paris Sér. I math., 307, 919-922, (1988) · Zbl 0657.60033
[4] Chung, K.L., ()
[5] Cox, J.T.; Grimmett, G., Central limit theorems for associated random variables and the percolation model, Ann. probab., 12, 514-528, (1984) · Zbl 0536.60094
[6] Dudley, R.M., Real analysis and probability, (1989), Wadsworth, Belmont, CA · Zbl 1023.60001
[7] Dudley, R.M.; Philipp, W., Invariance principles for Banach space valued random elements and empirical processes, Z. wahrsch. verw. gebiete, 62, 509-552, (1983) · Zbl 0488.60044
[8] Einmahl, U., Strong invariance principles for partial sums of independent random vectors, Annals probab., 15, 1419-1440, (1987) · Zbl 0637.60041
[9] Philipp, W., Invariance principles for sums of mixing random elements and the multivariate emprirical process, Colloq. math. soc. János bolyai, vol. 36, limit theorems in probability and statistics, 36, 843-873, (1982), Veszprém, Hungary
[10] Philipp, W., Invariance principles for independent and weakly dependent random variables, () · Zbl 0614.60027
[11] Philipp, W.; Stout, W., Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. amer. math. soc., 161, (1975) · Zbl 0361.60007
[12] Resnick, S.I., Extreme values, regular variation and point processes, (1987), Springer New York · Zbl 0633.60001
[13] Schatte, P., On strong versions of the central limit theorem, Math. nachr., 137, 249-256, (1988) · Zbl 0661.60031
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.