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The following first order asymptotics is well-known: $P(\|W\|_\infty < r)=\exp\{-\frac{\pi^2}{8 r^2}(1+o(1))\}$ as $r\to 0$, where $W$ is the Wiener process on [0,1].

Has anybody met an exact formula for $P(\|W\|_\infty < r)=?$ (for arbitrary $0<r<1$) or at least more precise asymptotics (the second or the third order)?

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See formula (7.15) on p.218 of Mörters-Peres "Brownian motion" (it is better suited for the case $r\to 0$ than (7.14) of Theorem 7.45 there).

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  • $\begingroup$ Is there similar exact formula for the centered Poisson process to stay within an interval? $\endgroup$
    – Elena Shmileva
    Commented Aug 30, 2016 at 17:15
  • $\begingroup$ Thank you, that's what was needed! It seems that the next order after $r^{−2}$ is a constant. $\endgroup$
    – Elena Shmileva
    Commented Aug 31, 2016 at 10:27
  • $\begingroup$ Не за что! For the centered Poisson process, I don't know... maybe, use the KMT strong approximation to be able to use the same formula?.. $\endgroup$
    – Serguei Popov
    Commented Aug 31, 2016 at 11:20

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