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Decimal expansion of lambda(2), a constant associated with the asymptotic upper tail of the distribution of the first hitting time T_{2,0} for an Ornstein-Uhlenbeck process across the level 2, starting at 0.
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%I #14 Jun 03 2017 15:33:26

%S 9,7,2,7,4,5,9,5,8,5,8,8,4,3,4,5,1,9,7,7,1,6,6,9,3,5,3,3,5,9,7,2,4,7,

%T 5,3,4,9,6,6,1,5,5,1,5,8,2,8,3,9,6,7,9,9,9,6,6,6,0,6,2,6,9,1,6,7,5,9,

%U 8,7,4,6,2,7,9,4,4,4,3,2,3,1,4,9,7,8,6,1,0,6,5,7,9,5,0,8,2,7,9,3,0,8,8,9

%N Decimal expansion of lambda(2), a constant associated with the asymptotic upper tail of the distribution of the first hitting time T_{2,0} for an Ornstein-Uhlenbeck process across the level 2, starting at 0.

%H Steven R. Finch, <a href="/A249417/a249417.pdf">Ornstein-Uhlenbeck Process</a>, May 15, 2004, p. 7. [Cached copy, with permission of the author]

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ParabolicCylinderFunction.html">Parabolic Cylinder Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process">Ornstein-Uhlenbeck process</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Parabolic_cylinder_function">Parabolic cylinder function</a>

%F lambda(a) = lim_{t->infinity} (1/t)*log(P(T_{a,0}>t)).

%F lambda(a) is the zero of D_{-lambda}(-a) closest to 0, where D_nu(x) is the parabolic cylinder function or Weber function.

%e -0.09727459585884345197716693533597247534966155...

%t lambda[a_?NumericQ] := x /. FindRoot[ParabolicCylinderD[-x, -a] == 0, {x, 0}, WorkingPrecision -> 104]; RealDigits[lambda[2]] // First

%Y Cf. A249417, A249418, A249445, A249449, A249451.

%K nonn,cons

%O -1,1

%A _Jean-François Alcover_, Nov 27 2014