Chernoff's distribution
In probability theory, Chernoff's distribution, named after Herman Chernoff, is the probability distribution of the random variable Z = argmax s ∈ R ( W ( s ) − s 2 ) , {\displaystyle Z={\underset {s\in \mathbf {R} }{\operatorname {argmax} }}\ (W(s)-s^{2}),} where W is a "two-sided" Wiener process (or two-sided "Brownian motion") satisfying W(0) = 0. If V ( a , c ) = argmax s ∈ R ( W ( s ) − c ( s − a ) 2 ) , {\displaystyle V(a,c)={\underset {s\in \mathbf {R} }{\operatorname {argmax} }}\ (W(s)-c(s-a)^{2}),} then V(0, c) has density f c ( t ) = 1 2 g c ( t ) g c ( − t ) {\displaystyle f_{c}(t)={\frac {1}{2}}g_{c}(t)g_{c}(-t)} where gc has Fourier transform given by g ^ c ( s ) = ( 2 / c ) 1 / 3 Ai ( i ( 2 c 2 ) − 1 / 3 s ) , s ∈ R {\displaystyle {\hat {g}}_{c}(s)={\frac {(2/c)^{1/3}}{\operatorname {Ai} (i(2c^{2})^{-1/3}s)}},\ \ \ s\in \mathbf {R} } and where Ai is the Airy function.