On the Uncertainty Relation for Positive Definite Probability Densities

Abstract

To each positive definite probability density there exists an adjoint density which is proportional to the characteristic function of p. The products have a greatest lower bound Λ, and it is known that 0.5276… < Λ ≤ 0.8609… We present a positive definite density with λ(p) = 6/7 and thereby improve the upper estimate. For densities representable as normal scale mixtures, we show that λ(p) ≥ 1 with equality if and only if p is a normal probability density.

AMS 1991 Subject Classification::

• Primary 60E05
• 60E10
• Secondary 42A82

Keywords:

• Dual density
• Laue's constant
• positive definite
• uncertainty principle
• normal scale mixture