Chernoff's distribution

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In probability theory, Chernoff's distribution, named after Herman Chernoff, is the probability distribution of the random variable

where W is a "two-sided" Wiener process (or two-sided "Brownian motion") satisfying W(0) = 0. If

then V(0, c) has density

where gc has Fourier transform given by

and where Ai is the Airy function. Thus fc is symmetric about 0 and the density ƒZ = ƒ1. Groeneboom (1989) shows that

where is the largest zero of the Airy function Ai and where .


  • Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields. 81: 79–109. doi:10.1007/BF00343738. MR 0981568.
  • Groeneboom, Piet; Wellner, Jon A. (2001). "Computing Chernoff's Distribution". Journal of Computational and Graphical Statistics. 10 (2): 388–400. CiteSeerX doi:10.1198/10618600152627997. MR 1939706.
  • Piet Groeneboom (1985). Estimating a monotone density. In: Le Cam, L.E., Olshen, R. A. (eds.), Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, vol. II, pp. 539–555. Wadsworth.