×

Strong limit theorems of Cesaro type means for arrays of orthogonal random elements with multi-dimensional indices in Banach spaces. (English) Zbl 1056.60027

Summary: The paper investigates the strong limit behavior of the Cesaro type average of arrays of James-type orthogonal random elements in Banach spaces. Particularly, it will be shown that the \(d\)-dimensional Banach spaces version condition \(\sum_n (E\|X_n\|^p/|n|^p)<\infty\) is sufficient to yield \[ \lim_{\min\limits_{l\leq i\leq d}(n_i)\to \infty} \frac{1}{|n|} \sum_{k\leq n}\;\prod_{l=1}^d \left(1-\frac{k_i-1}{n_i}\right) X_k=0 \text{ a.s}. \]

MSC:

60F15 Strong limit theorems
60B11 Probability theory on linear topological spaces
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
15B52 Random matrices (algebraic aspects)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Howell J.O., Lecture Notes in Math 860, in: Proceedings of the Conference on Probability in Banach Spaces pp 219– (1980) · doi:10.1007/BFb0090617
[2] DOI: 10.1016/0047-259X(89)90039-0 · Zbl 0685.60033 · doi:10.1016/0047-259X(89)90039-0
[3] Moricz F., Math. Nachricten 138 pp 243– (1988)
[4] Moricz F., Preprint, in: Strong Limit Theorems for p-Orthgonal Random Elements in Banach Spaces (1990)
[5] Moricz F., Acta Sci. Math. 55 pp 309– (1991)
[6] Revesz P., The Laws of the Large Numbers (1968) · Zbl 0155.01701
[7] Revesz P., Lecture Notes in Math. 672, in: Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces (1978)
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.