Bennett Eisenberg

Professor of Mathematics

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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.
• A Modification of Sylvester’s Four Point Problem, Math. Magazine, 2011.
• 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).