Bennett Eisenberg

Professor of Mathematics

Department of Mathematics #14
Lehigh University
Bethlehem, PA 18015
Telephone: Office: 610-758-3736 Home: 610-865-5289
e-mail: BE01@Lehigh.edu

Employment history:

Lehigh University 1972-present
University of New Mexico 1970-1972
Cornell University 1967-1970

Education:

Ph.D. 1968, M.I.T.
A.B. 1964, Dartmouth College
Coolidge High School, Washington, D.C., 1960

Research areas: Gaussian processes, ergodic theory, stationary processes, sequential statistical tests, epidemiology, record values, geometry and geometric probability, probability theory.

Selected publications categorized according to research area.

Gaussian processes:

The Relation of the Equivalence Conditions for Brownian Motion to the Equivalence Conditions for Certain Stationary Processes, Ann. Math. Stat. 1969.
Translating Gaussian Processes, Ann. Math. Stat. 1970.
Baxter's Theorem and Varberg's Conjecture. Pacific J. Math. 1972.
The Equivalence Singularity Problem for Gaussian Signals in Gaussian Noise. IEEE Trans. on Information Theory. 1986.

Ergodic theory:

Generalized Summing Sequences and the Mean Ergodic Theorem, Proc. Amer. Math. Soc. 1974 (with J. Blum).
Ergodic Theory and the Measure of Sets in the Bohr Group, Acta Sci. Math. (Szeged) 1973 (with J. Blum and L-S. Hahn).
Ergodic Theorems for Mixing Transformation Groups. Rocky Mt. J. of Math. 1979 (with J. Blum).

Stationary Processes:

Conditions for Metric Transitivity for Stationary Gaussian Processes on Groups. Ann. Math. Stat. 1972 (with J. Blum).
A Note on Metric Transitivity for Stationary Gaussian Processes on Groups. Ann. Math. Stat. 1972 (with J. Blum).
Linear Estimation of Regression Coefficients. Quarterly of Applied Math. 1974 (with R. Adenstedt).
The Law of Large Numbers for Subsequences of a Stationary Process. Annals of Prob. 1975 (with J. Blum).
A Note on Random Measures and Moving Averages on Non-Discrete Groups. Ann. Math. Stat. 1973 (with J. Blum).
Spectral Properties of Processes Derived from Stationary Gaussian Sequences. Stoch. Proc. and their Appl. 1974.
Prediction Theory and Ergodic Spectral Decompositions, Ann. of Prob. 1976 (with J. Blum).

Sequential statistical tests:

Non-Optimality of Likelihood Ratio Tests in the Sequential Detection of Signals in Gaussian Noise. Statistical Decision Theory. 1971.
Properties of Generalized Sequential Probability Ratio Tests. Ann. of Stat. 1976 (with B. Ghosh and G. Simons).
On Weak Admissibility of Tests. Ann. of Stat. 1978 (with G. Simons ).
The Likelihood Ratio and its Applications to Sequential Analysis. Ann. of Stat. 1980 (with B. Ghosh).
Curtailed and Uniformly Most Powerful Sequential Tests. Ann. of Stat. 1980 (with B. Ghosh).
On the Sample Size of Curtailed Tests. Comm. in Stat.-Theory and Methods. 1981 (with B. Ghosh).
The Asymptotic Solution of the Kiefer-Weiss Problem. Sequential Analysis. 1982.
Multihypothesis Problems. Handbook of Sequential Analysis. 1991.
The Sequential Probability Ratio Test. Handbook of Sequential Analysis. 1991.
One-sided SPRT's which Hit the Boundary. Sequential Analysis. 1995
Average Sample Number, Kluwer Encyclopedia of Mathematics, 2002.
Sequential Probability Ratio Test, Kluwer Encyclopedia of Mathematics, 2002.
Discussion on "Likelihood Ratio Identities and Their Application in Sequential Analysis" by T.L. Lai, Sequential Analysis, 2004.
Discussion on "Life and Work of Bhaskar Kumar Ghosh" by Pranab Kumar Sen, Sequential Analysis, 2010 (with W. Huang).

Epidemiology.

The Number of Partners and the Probability of HIV Infection. Stat. in Medicine. 1989.
The Effect of Variable Infectivity on the Risk of HIV Infection. Stat. in Medicine. 1990.

Record Values.

The Asymptotic Probability of a Tie for First Place. Ann. of Applied Prob. 1993 (with G. Strang and G. Stengle).
A Necessary and Sufficient Condition for the Existence of the Limiting Probability of a Tie for First Place. Stat. and Prob. Letters. 1995 (with Y. Baryshnikov and G. Stengle).
Minimizing the Probability of a Tie for First Place. J. of Math. Analysis and its Applic. 1996 (with G. Stengle).
On the Expectation of the Maximum of I I D Geometric Random Variables, Stat. and Prob. Letters. 2008.
The Number of Players Tied for the Record, Stat. and Prob. Letters. 2009.

Geometry and Geometric Probability

Optimal Locations. College Math. J. 1992 (with S. Khabbaz).
Random Triangles in n-Dimensions. Amer. Math. Monthly. 1996 (with R. Sullivan).
Crofton's Differential Equation. Amer. Math. Monthly. 2000 (with R. Sullivan).
The Fundamental Theorem of Calculus in Two Dimensions, Amer. Math. Monthly. 2002 (with R. Sullivan).
Surfaces of Revolution in Four Dimensions, Math. Magazine, 2004.

Probability Theory.

Another Look at Korovkin's Theorem. J. of Approx. Theory. 1976.
Uniform Convergence of Distribution Functions. Proc. Amer. Math. Soc. 1983 (with Shixin Gan).
Positive Martingales and their Induced Measures. Proc. Amer. Math. Soc. 1983 (with Shixin Gan).
Independent Events in a Discrete Uniform Probability Space. The Amer. Statistician. 1987 (with B. Ghosh).
A Variation of a Theorem of Sparre-Andersen, Stat. and Prob. Letters. 1992.
Independent Variables with Independent Sum and Difference. J. of Multivariate Anal. 1993 (with Y. Baryshnikov and W. Stadje).
Independent Events and Independent Experiments. Proc. Amer. Math.Soc. 1993 (with Y. Baryshnikov).
What is the Margin of Error of a Poll? College Math. J. 1997
How Much Should You Pay for a Derivative? College Math. J. 1998.
Generalizations of Markov's Inequality, Stat. and Prob. Letters. 2001 (with B. Ghosh).
Why is the Sum of Independent Normal Random Variables Normal? Math. Magazine, 2008 (with R. Sullivan).