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A liminf result in Strassen’s law of the iterated logarithm. (English) Zbl 0719.60033

Limit theorems in probability and statistics, Proc. 3rd Colloq., Pécs/Hung. 1989, Colloq. Math. Soc. János Bolyai 57, 83-93 (1990).
[For the entire collection see Zbl 0713.00013.]
Let \(\{\) W(t), \(t\geq 0\}\) be a standard Wiener process and put \[ X_ t(s)=W(st)/(2t \log \log t)^{1/2},\quad 0\leq s\leq 1. \] Let \(f(s)=as^ 2/2+bs\) be a function from the compact set in Strassen’s law of the iterated logarithm. Assume that \(\int^{1}_{0}f'(s)^ 2ds=1\). It is shown that \[ \liminf_{t\to \infty}(\log \log t)^{2/3}\| X_ t- f\| =(\mu_ 0/4)^{1/3}\quad a.s., \] where \(\| \|\) denotes the sup-norm in C[0,1] and \(\mu =\mu_ 0\) is the smallest positive eigenvalue of the problem \[ y''+\mu (| a| x+| a+b|)y=0,\quad y(-1)=y(1)=0. \] In the particular case when \(a=0\), \(b=1\), we have \(\mu_ 0=\pi^ 2/8\).

MSC:

60F15 Strong limit theorems
60J65 Brownian motion

Citations:

Zbl 0713.00013