A249451 Decimal expansion of lambda(1), a constant associated with the asymptotic upper tail of the distribution of the first hitting time T_{1,0} for an Ornstein-Uhlenbeck process across the level 1, starting at 0.

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

Offset

0,1

Comments

Following Steven Finch, it is assumed that the values of the parameters of the stochastic differential equation dX_t = -rho (X_t - mu) dt + sigma dW_t, satisfied by the process, are mu = 0, rho = 1 and sigma^2 = 2.

Links

Table of n, a(n) for n=0..99.

Steven R. Finch, Ornstein-Uhlenbeck Process, May 15, 2004, p. 7. [Cached copy, with permission of the author]

Eric Weisstein's MathWorld, Parabolic Cylinder Function

Wikipedia, Ornstein-Uhlenbeck process

Wikipedia, Parabolic cylinder function

Formula

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

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

Example

-0.38823829470678552935396415444674775420950403550518599...

Mathematica

lambda[a_?NumericQ] := x /. FindRoot[ParabolicCylinderD[-x, -a] == 0, {x, 0}, WorkingPrecision -> 100]; RealDigits[lambda[1]] // First

Crossrefs

Cf. A249417, A249418, A249445, A249449.

Sequence in context: A101749 A206431 A301484 * A019744 A252876 A102639

Adjacent sequences: A249448 A249449 A249450 * A249452 A249453 A249454

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 29 2014

Status

approved