Low-discrepancy Sequence - Definition of Discrepancy

Definition of Discrepancy

The discrepancy of a set P = {x1, ..., xN} is defined, using Niederreiter's notation, as

D_N(P) = \sup_{B\in J} \left| \frac{A(B;P)}{N} - \lambda_s(B) \right|

where λs is the s-dimensional Lebesgue measure, A(B;P) is the number of points in P that fall into B, and J is the set of s-dimensional intervals or boxes of the form

where .

The star-discrepancy D*N(P) is defined similarly, except that the supremum is taken over the set J* of intervals of the form

where ui is in the half-open interval [0, 1).

The two are related by

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