Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Deduce from Theorem 12.5 the following test for upper functions in large time. Assume that \(\kappa \in \mathcal{C}[1, \infty)\) is a positive function such that \(\kappa(t) / t\) is decreasing and \(\kappa(t) / \sqrt{t}\) is increasing. Then \[\mathbb{P}(B(t)

Data From Theorem 12.5

 

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Ex. 12.4 12.5 Theorem (Kolmogorov's test). Let (B)o be a BMT and x: (0, 1) (0,00) continuous function such that it) is increasing and sit VT is decreasing. Then the Ex.12.5 following alternative holds: (1)-1 P(1)-1. D) < ()() (12.6) (12.7)