n,m,a,b] properties of modulus of continuity, use Cauchy-Schwartz inequality and (4) , (11) we have
Now use properties of modulus of continuity then we have
X] with modulus of continuity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
9 If X is bounded and bounds for the density fx and its modulus of continuity are known explicitly, the last result is strong enough to allow, in principle, perfect simulation using von Neumann's rejection method; see Devroye (2001) for the case of infinitely divisible perpetuities with approximation of densities by Fourier inversion, Devroye, Fill, and Neininger (2000) for the case of the Quicksort limit distribution and Devroye and Neininger (2002) for more general fixed-point equations of type (2).
Again the inf in the tail term ensures that if f is a polynomial of degree [less than or equal to] r - 1, then the modulus of continuity
vanishes identically, as is expected from an rth order modulus.
For f [member of] C[0, 1], [delta] > 0, we define the modulus of continuity
[omega](f, [delta]) as follows: