Inequalities and bounds for variances of point processes and fibre processes

Abstract

For the variances of numbers of points and total fibre lengths in compact convex sets, corresponding to point processes and fibre processes, bounds are proved. They base on monotonicity and convexity properties of a function of geometrical nature occuring in the formula for the variance when using the reduced second moment measure. As particular cases, stationary renewal processes. Cox processes and Neyman-Scott processes are considered. For certain fibre process models Poisson point processes and segment processes are used as approximative models.

Keywords:

• Point process
• fibre process
• variance
• reduced second moment measure
• partial ordering
• bound
• renewal process
• segment process