ON RESCALED POISSON PROCESSES AND THE BROWNIAN BRIDGE

FREDERIC PAIK SCHOENBERG

Department of Statistics, University of California, Los Angeles, Los Angeles, CA 90095–1554, U.S.A., e-mail: frederic@stat.ucla.edu

(Received April 13, 2000; revised October 10, 2000)

Abstract. The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian bridge are explored, and implications for point process residual analysis are discussed.

Key words and phrases: Brownian bridge, Poisson bridge, intensity, point process, Poisson process, residual analysis.