Summary
Let β T = (2a T(log Ta −1T +log log T))−1/2, 0< a T ≦T<∞ and let R* be the set of sub-rectangles of the square [0, T 1/2]x[0, T1/2], having an area a T . This paper studies the almost sure limiting behaviour of \(\beta _T \mathop {\sup }\limits_{R \in R*} |W(R)|\) as T→∞, where W is a two-time parameter Wiener process. With a T =T, our results give the well-known law of iterated logarithm and a generalization of the latter is also attained. The multi-time parameter analogues of our twotime parameter Wiener process results are also stated in the text.
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Research partially supported by a Canadian NRC grant and a Canada Council Leave Fellowship
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Csörgö, M., Révész, P. How big are the increments of a multi-parameter Wiener process?. Z. Wahrscheinlichkeitstheorie verw Gebiete 42, 1–12 (1978). https://doi.org/10.1007/BF00534203
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DOI: https://doi.org/10.1007/BF00534203