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  1. Probabilizing Fibonacci Numbers P. Diaconis. (2017)
    Submitted [PDF]


  1. A Central Limit Theorem for a New Statistic on Permutations. S. Chatterjee & P. Diaconis. (2016)

  2. A. Hurwitz and the Origins of Random Matrix Theory in Mathematics. P. Diaconis & P.J. Forrester. (2016).
    Submitted [PDF]

  3. An Exercise (?) in Fourier Analysis on the Heisenberg Group. D. Bump, P. Diaconis, A. Hicks, L. Miclo & H. Widom. (2016).
    To appear: Ann. Fac. Sci. Toulouse Math. arXiv

  4. Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks. L. Miclo. & P. Diaconis. (2016).
    Science China Mathematics 59(2):205-226 (Feb.)
    DOI 10.1007/s11425-015-5085-2 Springer [PDF]

  5. Probabilizing Parking Functions. P. Diaconis & A. Hicks. (2016).
    Submitted. [PDF]


  1. Central Limit Theorems for Some Set Partition Statistics. B. Chern, P. Diaconis, D. Kane & R.C. Rhoades. (2015).
    Advances in Applied Math 70:92-105.
    DOI:10.1016/j.aam.2015.06.008 Science Direct [PDF]

  2. de Finetti Priors using Markov chain Monte Carlo computations. S. Bacallado, P. Diaconis & S. Holmes. (2015).
    Statistics and Computing, 25 4:797-808.(July)
    DOI:10.1007/s11222-015-9562-9 Springer [PDF]

  3. Random Walk on Unipotent Matrix Group. P. Diaconis & R. Hough. (2015).

  4. The Sample Size Required in Importance Sampling. S. Chatterjee & P. Diaconis. (2015).
    Submitted. [PDF]

  5. A Spectral Analysis Approach for Experimental Designs. R.A. Bailey, P. Diaconis, D.N. Rockmore & C. Rowley. (2015).
    Excursions in Harmonic Analysis, Volume 4: The February Fourier Talks at the Norbert Wiener Center. J. Benedetto (ed) pp367-395. Springer International Publishing DOI 10.1007/978-3-319-20188-7_14
    Springer [PDF]

  6. Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper's Operators.
    D. Bump, P. Diaconis, A. Hicks, L. Miclo & H. Widom. (2015).
    To appear: Jour. of Operator Theory [PDF]



  1. Carries, Group Theory and Additive Combinatorics. P. Diaconis, X. Shao & K. Soundararajan. (2014).
    Amer. Math. Monthly, 121(8):674-688. (Oct.) DOI: 10.4169/amermathmont.121.08.674
    MAA link [PDF]

  2. Closed Expressions for Averages of Set Partition Statistics. B. Chern, P. Diaconis, D. Kane & R. Rhoades. (2014).
    Research in the Math. Sci. 1:2 doi:10.1186/2197-9847-1-2
    Springer Open Journal [PDF]

  3. Combinatorics of Balanced Carries. P. Diaconis & J. Fulman. (2014).
    Adv. Appl. Math. 59:8-25
    DOI: 10.1016/j.aam.2014.05.005 Science Direct [PDF]

  4. Convolution powers of complex functions on Z. P. Diaconis & L. Saloff-Coste. (2014).
    Mathematische Nachrichten, 287(10):1106-1130. (Feb.) DOI: 10.1002/mana.201200163
    Wiley link [PDF]

  5. Five stories for Richard. P. Diaconis. (2014).

  6. Fluctuations of the Bose-Einstein Condensate. S. Chatterjee & P. Diaconis. (2014).
    Jour. of Physics A: Mathematical and Theoretical 47(8) 085201 DOI: 10.1088/1751-8113/47/8/08521.
    IOP Science link [PDF]

  7. Hopf algebras and Markov chains: Two examples and a theory. P. Diaconis, A. Pang & A. Ram. (2014).
    Jour. of Alg. Comb. 39:(3)527-585 [PDF] arXiv link

  8. An Introduction to Multivariate Krawtchouck Polynomials and their Applications. P. Diaconis & R. Griffiths. (2014).
    Jour. Stat. Planning and Inference., 154:39-53. (Nov.)
    Science Direct [PDF]

  9. The Mathematics of the Flip and Horseshoe Shuffles. S. Butler, P. Diaconis & R. Graham. (2014).
    submitted [PDF]

  10. On quantitative convergence to quasi-stationarity. P. Diaconis & L. Miclo. (2014).
    submitted [PDF]

  11. Unseparated Pairs and Fixed Points in Random Permutations. P. Diaconis, S. Evans & R. Graham. (2014).
    Adv. Appl. Math. 61:102-124. (Oct.) doi:10.1016/j.aam.2014.05.006 Science Direct [PDF]



  1. Analysis of Casino Shelf Shuffling Machines. P. Diaconis, J. Fulman & S. Holmes. (2013).
    Ann. Appl. Probab. 23(4):1692-1720.

  2. Estimating and Understanding Exponential Random Graph Models. P. Diaconis & S. Chatterjee (2013).
    Ann. Statist., 41(5):2428-2461. [PDF]

  3. Interval Graph Limits. P. Diaconis, S. Holmes & S. Janson. (2013).
    Annals of Combinatorics, 17(1):27-52

  4. Note on a partition limit theorem for rank and crank. P. Diaconis, S. Janson & R. Rhoades. (2013).
    Bull. London Math. Soc., 45(3):551-553. DOI: 10.1112/blms/bds121 arXiv link

  5. Some things we've learned (about Markov chain Monte Carlo). P. Diaconis. (2013).
    Bernoulli 19(4):1294-1305 DOI: 10.3150/12-BEJSP09 [PDF]

  6. Universal Limit Theorems in Graph Coloring Problems with Connections to Extremal Combinatorics.
    B. Bhattacharya, P. Diaconis & S. Mukherjee. (2013).
    submitted [PDF]



  1. Exchangeable Pairs of Bernoulli Random Variables, Krawtchouck Polynomials, and Ehrenfest Urns.
    P. Diaconis & R. Griffiths. (2012).
    Australian & New Zealand Journal of Statistics, 54(1):81-101. DOI: 10.1111/j.1467-842X.2012.00654.x

  2. Foulkes Characters, Eulerian Idempotents, and An Amazing Matrix. P. Diaconis & J. Fulman. (2012).
    Jour. Alg. Comb. 36(3):425-440. [PDF]

  3. On Dirichlet eigenvectors for neutral two-dimensional Markov chains. N. Champagnat, P. Diaconis & L. Miclo. (2012).
    E. Jour. Prob. 17 article 63 [PDF] DOI: 10.1214/EJP.v17-1830

  4. A probabilistic interpretation of the Macdonald polynomials. P. Diaconis & A. Ram. (2012).
    Ann. Appl. Prob. 40(5):1861-1896. DOI: 10.1214/11-AOP674 [PDF]

  5. On the spectral analysis of second-order Markov chains. P. Diaconis & L. Miclo. (2012).
    Ann. Fac. Sci. Toulouse Math 22(3):573-621 [PDF]

  6. Riffle Shuffles with Biased Cuts. S. Assaf, P. Diaconis & K. Soundararajan. (2012).
    DMTCS proceedings, FPSAC 2012, Nagoya, Japan.
    To appear: Mathematische Nachirichton [PDF]

  7. Random doubly stochastic tridiagonal matrices. P. Diaconis & P. Wood. (2012).
    Random Structures & Algorithms, 42(4):403-437.
    DOI: 10.1002/rsa.20452 [PDF]

  8. Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. P. Diaconis, M. Isaacs, et al. (2012). Advances in Mathematics 229(4):2310-2337.



  1. Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks by P. Diaconis & R. Graham.
    Princeton University Press. (October 23, 2011) ISBN-10: 0691151644

  2. Gibbs/Metropolis algorithm on a convex polytope. P. Diaconis, G. Lebeau & L. Michel. Math Zeitschrift,
    published online: Aug. 2011. DOI: 10.1111/j.1467-842X.2012.00654.x [PDF]

  3. The mathematics of mixing things up. P. Diaconis. (2011). Journal of Statistical Physics 144(3):445-459. [PDF]

  4. On Barycentric Subdivision. P. Diaconis & L. Miclo. (2011). Combinatorics, Probability and Computing, 20(2):213-237.
    published online: 24 Nov. 2010. [PDF] DOI:10.1017/S0963548310000441

  5. Random Graphs with a Given Degree Sequence. P. Diaconis, S. Chatterjee & A. Sly. (2011). Ann. Appl. Prob., 21(4):1400-1435 arXiv:1005.1136v3 [PDF]

  6. Sampling from a Manifold. P. Diaconis, S. Holmes & M. Shahashahani. (2011).
    Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 10:102-125 [PDF]



  1. Functions of Random Walks on Hyperplane Arrangements. P. Diaconis & C.A. Athanasiadis. (2010).
    Adv. in Appl. Math. 45(3):410-437, arXiv:0912.1786v1 [PDF]

  2. Properties of Uniform Doubly Stochastic Matrices. P. Diaconis, S. Chatterjee & A. Sly. (2010).
    To appear: Ann. Appl. Prob. arXiv link [PDF]

  3. Ajout de nombres, et processus des retenues. P. Diaconis. (2010). [PDF]

  4. Gibbs Sampling, Conjugate Priors and Coupling. P. Diaconis, K. Khare & L. Saloff-Coste. (2010).
    Sankhya 72-A part 1:136-169 [PDF]

  5. Geometric Analysis for the Metropolis Algorithm on Lipschitz Domains. P. Diaconis, G. Lebeau & L. Michel. (2010).
    Invent. Math. 185(2):239-281. DOI 10.1007/s00222-010-0303-6 [PDF]

  6. Stochastic alternating projections. Diaconis, P.,Khare, K. & Saloff-Coste, L. (2010).
    Illinois Journal of Mathematics 54(3):963-979.



  1. On Characterizations of Metropolis type Algorithms in continuous time. P. Diaconis & L. Miclo. (2009).
    Alea 6:199-238. [PDF]

  2. Book Review: Probabilistic Symmetries and Invariance Principles
    by Olav Kallenberg,
    Probability and its Applications, Springer, NY. Bulletin (New Series) of the American Mathematical Society. April 2009.
    DOI:10.1090/S0273-0979-09-01262-2 [PDF]

  3. Carries, Shuffling and an Amazing Matrix. P. Diaconis & J. Fulman. (2009).
    Amer. Math. Mo., 116(9):788-803. DOI: 10.4169/000298909X474864 [PDF]

  4. Riffle shuffles of a deck with repeated cards. P. Diaconis Assaf, S. & Soundararajan, K. (2009).
    DMTCS Proceedings, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 0(1):89-102. [PDF]

  5. Carries, Shuffling and Symmetric Functions. P. Diaconis & J. Fulman. (2009). Adv. in Appl. Math., 43(2):176-196. doi:10.1016/j.aam.2009.02.002 [PDF]

  6. Fastest Mixing Markov Chains on Graphs with Symmetries. P. Diaconis, S. Boyd, P. Parrilo & L. Xiao. (2009).
    SIAM J. Optim., 20(2):792-819. doi- [link]

  7. Micro-local Analysis for the Metropolis Algorithm. P. Diaconis & G. Lebeau. (2009).
    Mathematische Zeitschrift, 262(2):441-447. [PDF]

  8. On adding a list of numbers (and other one-dependent determinantal processes). P. Diaconis, A. Borodin & J. Fulman. (2009).
    Bulletin (New Series) of the Amer. Math. Soc., 47(4):639-670 [PDF]

  9. On Times to Quasi-Stationary for Birth and Death Processes.
    P. Diaconis & L. Miclo. (2009). Journal of Theoretical Probability, 22(3):558-586. [PDF]

  10. A Sequential Importance Sampling Algorithm for Generating Random Graphs With Prescribed Degrees.
    P. Diaconis & J. Blitzstein. (2009). Journal of Internet Mathematics 6(4):489-522 [PDF]

  11. Supercharacter Formulas for Pattern Groups. P. Diaconis & N. Thiem. (2009).
    Trans. Amer. Math. Soc., 361:3501-3533. [PDF]

  12. Threads Through Group Theory. Character Theory of Finite Groups - Conference on Character Theory of Finite Groups,
    held at the Universitat de València, Spain, on June 3-5, 2009, in honor of I. Martin Isaacs. Contemporary Mathematics, Vol. 524, p33-45. AMS, Providence, RI, USA (2009). ISBN-10: 0-8218-4827-5 [PDF]

  13. Threshold Graph Limits and Random Threshold Graphs.
    P. Diaconis, S. Holmes & S. Janson. (2009). Journal of Internet Mathematics, 5(3):267-320.
    arXiv:0908.2448v1 [math.CO] [PDF]



  1. A Rule of Thumb for Riffle Shuffling. P. Diaconis, Assaf, S. & Soundararajan, K. (2008).
    Annals of Applied Probability 21(3):843-875. arXiv:0908.3462v1 [PDF]

  2. Gibbs sampling, exponential families and orthogonal polynomials.
    P. Diaconis, K. Khare & L. Saloff-Coste. (2008). Statistical Science, 23(2):151-178. DOI: 10.1214/07-STS252. [PDF]

  3. Comment. P. Diaconis & E. Lehmann. (2008). JASA 103(481):16 PDF

  4. Graph Limits and Exchangeable Random Graphs. P. Diaconis S. Janson. (2008). Rendiconti di Matematica, Serie VII 28:33-61. [PDF]

  5. Products of Universal Cycles; P. Diaconis & R.L. Graham. A lifetime of puzzles: a collection of puzzles in honor of Martin Gardner, Edited by E. Demain, M. Demaine & T. Rodgers. (2008). AK Peters, Ltd. Wellesley, MA. pp35-55 [PDF]

  6. The Markov Chain Monte Carlo Revolution. P. Diaconis. (2008). Bull. Amer. Math. Soc., Nov. 2008. DOI:10.1090/S0273-0979-08-01238-X [PDF]

  7. Projection Pursuit for Discrete Data. P. Diaconis & J. Salzman. (2008). Probability and Statistics: Essays in Honor of David A. Freedman. IMS Collections, Beachwood, Ohio, USA: Institute of Mathematical Statistics. pp. 265-288. [PDF]

  8. Supercharacters and Superclasses for Algebra Groups; P. Diaconis & I.M. Isaacs. (2008).
    Trans. Amer. Math. Soc., 360(5):2359-2392. MR 2373317 [PDF]



  1. On Fixed Points of Permutation. P. Diaconis. J. Fulman & R. Guralnick. (2007).
    Journal of Algebraic Combinatorics, 28:189-218 [PDF]

  2. Horseshoes in Multidimensional Scaling and Kernel Methods. P. Diaconis, S. Goel & S. Holmes. (2007). Annals of Applied Statistics, 2(3):777-807. [PDF]

  3. Hit and Run as a Unifying Device. P. Diaconis & H. Andersen. (2007). Journal de la Société Française de Statistique, 148(4):5-28. [PDF]

  4. Dynamical Bias in the Coin Toss. P. Diaconis, S. Holmes & R. Montgomery. (2007). SIAM Review, 49(2):211-235. [PDF]

  5. The Solutions to Elmsley's Problem. P. Diaconis & R.L. Graham. (2007). Math Horizons 14: 22-27. [PDF]



  1. Bayesian Analysis for Reversible Markov Chains. P. Diaconis & S. Rolles. (2006). Annals of Statistics, 34(3):1270-1292 [PDF]

  2. Examples Comparing Importance Sampling and the Metropolis Algorithm; P. Diaconis & F. Bassetti. (2006). Illinois Journal of Mathematics, 50(1-4):67-91 [PDF]

  3. Fastest Mixing of Markov Chain on a Graph and a Connection to a Maximum Variance Unfolding Problem. P. Diaconis, J. Sun, S. Boyd & L. Xiao. (2006). SIAM Review, 48(4):681-699. [PDF]

  4. Fastest Mixing Markov Chain on a Path. P. Diaconis, S. Boyd & L. Xiao. (2006). Amer. Math. Mo., 113(1):70-74. [PDF]

  5. Markov Bases for Noncommutative Fourier Analysis of Ranked Data. P. Diaconis N. Eriksson. (2006). Journal of Symbolic Computation, 41(2):182-195 [PDF]

  6. Mathematical Statistics. Princeton Companion for Mathematics. Edited by T. Gowers. Princeton University Press (2006).



  1. Analysis of a Bose-Einstein Markov Chain. P. Diaconis. (2005). Annales de l'Institut Henri Poincare, Prob. and Stat., 41(3):409-418. [PDF]

  2. Exchangeable Pairs and Poisson Approximation. P. Diaconis, S. Chatterjee & E. Meckes. (2005). Probability Surveys, 2(1):64-106. [PDF] [PostScript]

  3. Separation Cut-Offs for Death and Birth Chains; P. Diaconis & L. Saloff-Coste. (2005). Ann. of Appl. Prob. 16(4):2098-2122. [PDF]

  4. Sequential Monte Carlo Methods for Statistical Analysis of Tables. P. Diaconis, Y. Chen, S. Holmes & J.S. Liu. (2005). JASA, 100:109-120. [PDF]

  5. Solitaire: Man versus Machine. P. Diaconis, X. Yan, P. Rusmeuichientong, & B. van Roy. (2005). Management Science and Engineering Stanford Technical Reports Collection, Stanford University. [PDF]

  6. Symmetry Analysis of Reversible Markov Chains. P. Diaconis S. Boyd, P. Parrilo, & L. Xiao. (2005). Journal of Internet Mathematics, 2(1):31-71. [PDF]

  7. What is...a Random Matrix? P. Diaconis (2005). Notices of the AMS, 52(11):1348-1349. [PDF]



  1. Fastest Mixing Markov Chain on a Graph. P. Diaconis S. Boyd & L. Xiao. (2004). SIAM Review, 46(4):667-689. [PDF]

  2. Random Matrices, Magic Squares and Matching Polynomials. P. Diaconis & A. Gamburd. (2004). Electronic Journal of Combinatorics, 11(2) [PDF]

  3. Random Walks on Finite Groups, Probability on Discrete Structures Volume 110 of the series Encyclopaedia of Mathematical Sciences pp 263-346 [PDF]
    Springer link

  4. A Super-Class Walk on Upper-Triangular Matrices. P. Diaconis E. Arias-Castro & R. Stanley. (2004). Journal of Algebra., 278(2):739-765 [PDF]

  5. Uniqueness of the Invariant Distributions for Split-Merge Transformations and the Poisson-Dirichlet Law. P. Diaconis, E. Mayer-Wolf, O. Zeitouni & M. Zerner. (2004). Ann. of Prob. 32(1B):915-938. [PDF] [PostScript]

  6. Numerical Results for the Metropolis Algorithm. P. Diaconis J. Neuberger. (2004). Experimental Math.,13(2):207-214 [PDF]

  7. On the Distribution of the Greatest Common Divisor. P. Diaconis, P. Erdös & A. Dasgupta, ed., A Festschrift for Herman Rubin (2004). Beachwood, OH USA: Institute of Mathematical Statistics, 45:126-137.

  8. Stein's Method for Markov Chains: First Examples. P. Diaconis & S. Holmes, eds., (2004). Stein's Method: Expository Lectures and Applications 27-43. Beachwood, Ohio, USA: Institute of Mathematical Statistics

  9. Uses of Exchangeable Pairs in Monte Carlo Markov Chains. P. Diaconis, C. Stein, S. Holmes & G. Reinert. P. Diaconis and S. Holmes (eds.), Stein's Method: Expository Lectures and Applications, 1-26. (2004). Beachwood, Ohio, USA: Institute of Mathematical Statistics.

  10. The Markov Moment Problem and de Finetti's Theorem Part I. P. Diaconis & D. Freedman. (2004). Mathematische Zeitschrift, 247(1):183-199. [PDF]

  11. The Markov Moment Problem and de Finetti's Theorem Part II. P. Diaconis & D. Freedman. (2004). Mathematische Zeitschrift, 247(1):201-212. [PDF]



  1. Brownian Motion and the Classical Groups; P. Diaconis, A. D'Aristotile, C. Newman. K. Athreya, M. Majumdar, M. Puri, & E. Waymire (eds.). (2003). Probability, Statisitics and their applications: Papers in Honor of Rabii Bhattacharaya. pp 97-116. Beachwood, OH, USA: Institute of Mathematical Statistics

  2. Mathematical Developments from the Analysis of Riffle-Shuffling. In A. Ivanov, A. Fuanou & M. Liebeck (eds.), Groups, Combinatorics and Geometry, pp.73-97, World Scientific, N.J. (2003). [PDF]

  3. The Problem of Thinking Too Much. P. Diaconis. Bull Amer. Acad. Sci., Spring 2003, 26-38. [PDF] 1-26

  4. Random Walk on Groups: Characters and Geometry. C.M. Campbell, et al (eds).Groups St. Andrews, 2001. Oxford: Cambridge University Press, 1:120-142, (2003). [PDF]



  1. A Different Construction of Gaussian Fields From Markov Chains: Dirichlet Covariances. P. Diaconis & S. Evans. Ann. Inst. Henri Poincare; Prob. and Stat, 38(6):863-878, Nov.-Dec. 2002. DOI:10.1016/S0246-0203(02)01123-8 [PDF]

  2. G.H. Hardy and Probability??? P. Diaconis. (2002). Bull. London Math. Soc., 34(4):385-402 part 4. DOI: 10.1112/S002460930200111X. [PDF]

  3. Patterns in Eigenvalues: the 70th Josiah Gibbs Lecture. Bull. Amer. Math. Soc., 40(2):155-178, 2003. [PDF]

  4. Random Walk on Trees and Matchings. P. Diaconis & S. Holmes. Electronic Jour. Probab. 7 Paper 6, 1-17. [PDF]

  5. Unitary Correlations and the Fej$eacute;r Kernel. P. Diaconis, D. Bump & J. Keller. (2002). Mathematical Physics, Analysis and Geometry, 5(2):101-123. DOI: 10.1023/A:1016200519958 [PDF] [PostScript]

  6. The Asymmetric One-Dimensional Constrained Ising Model: Rigorous Results. P. Diaconis & D. Aldous. (2002). Jour. Statist. Physics. 107(5-6):945-975. DOI: 10.1023/A:1015170205728 [PDF] [PostScript]

  7. Mysteries of Cardano, the Probabilist. To appear in G. Cardano: Liber de Ludo Aleae. Gambacorta, ed.

  8. Toeplitz Minors. P. Diaconis & D. Bump. (2002). Jour. Combin. Th., (A) 97(2):252-271 [PDF]



  1. Statistical problems involving permutations with restricted positions. P. Diaconis, R.L. Graham & S. Holmes. Gunst, C. Klaassen, & A van der Vaart (eds.), State of the art in probability and statistics: Festschrift for Willem R. van Zwet, Papers from the symposium
    held at the University of Leiden, Leiden, March 23-26, 1999 (Beachwood, OH: Institute of Mathematical Statistics, 2001) [PDF] [PostScript]

  2. A Geometric Interpretation of the Metropolis-Hastings Algorithm. P. Diaconis & L. Billera. (2001). Statis. Sci., 16(4):335-339 [PDF] [PostScript]

  3. Linear Functionals of Eigenvalues of Random Matrices. P. Diaconis & S. Evans. (2001). Trans. of Amer. Math. Soc., 353(7):2615-33. [PDF]

  4. Combinatorics for the East Model. P. Diaconis, R.L. Graham & F. Chung. (2001). Adv. in Appl. Math., 27(1):192-206. [PDF]

  5. Chutes and Ladders in Markov Chains. P. Diaconis & R. Durret. (2001). Jour. Th. Prob., 14(3):899-926. [PDF]



  1. Bounds for Kac's Master Equation. P. Diaconis & L. Saloff-Coste. (2000). Communications Math. Phys., 209(3):729-55. [PDF]

  2. Analysis of a Nonreversible Markov Chain Sampler. P. Diaconis, S. Holmes & R. Neal. (2000). Ann. Appl. Probab., 10(3):726-52. [PostScript]

  3. Immanants and Finite Point Processes. P. Diaconis & S. Evans. (2000). Jour. Combin. Th. Series A, 91(1-2):305-321. [PDF] [PostScript]

  4. A Bayesian Peek into Feller I. P. Diaconis & S. Holmes. (2002). Sankhya A, 64(3):820-841. [PDF]

  5. Analysis of Systematic Scan Metropolis Algorithms Using Iwahori-Hecke Algebra Techniques. P. Diaconis & A. Ram, Michigan Journal of Mathematics, 48(1):157-190. [PDF]



  1. Random Walks and Plane Arrangements in Three Dimensions. P. Diaconis, L. Billera & K. Brown. (1999). Amer. Math. Monthly, 106(6):502-24. [PDF]

  2. Longest Increasing Subsequences: From Patience Sorting to the Baik-Dieft-Johansson Theorem. P. Diaconis & D. Aldous. (1999). Bull. Amer. Math. Soc., 36:413-32.[PDF] [PostScript]

  3. Iterated Random Functions. P. Diaconis D. Freedman (1999). SIAM Review, 41(1):45-76. [PDF]

  4. New Tests of the Correspondence Between Unitary Eigenvalues and the Zeros of Riemann's Zeta Functions. P. Diaconis & M. Coram. (1999). Jour. Physics A: Math. and General, 36(12):2883-2906. [PDF]



  1. Random Walk and Hyperplane Arrangements. P. Diaconis & K. Brown. Ann. Probab., 26(4):1813-54. [PDF]

  2. Algebraic Algorithms for Sampling from Conditional Distributions. P. Diaconis & B. Sturmfels. Ann. Statist., 26(1):363-97. [PDF]

  3. Lattice Walks and Primary Decomposition. P. Diaconis, D. Eisenbud & B. Sturmfels, Mathematical Essays in Honor of Gian-Carlo Rota, pp 173-94. B. Sagan & R.P. Stanley (eds.),

  4. Consistency of Bayes Estimates for Nonparametric Regression: Normal Theory. P. Diaconis & D. Freedman, Bernoulli, 4(4):411-444. [PDF]

  5. What Do We Know About the Metropolis Algorithm? P. Diaconis & L. Saloff-Coste, Jour. Comp. System Sciences, 57:20-36. [PDF]

  6. Walks on Generating Sets of Groups. P. Diaconis & L. Saloff-Coste, Inventiones Math., 134(2):251-300. [PDF]

  7. The Graph of Generating Sets of an Abelian Group. P. Diaconis & R. L. Graham, Colloquium Math., 31-8. [PDF]

  8. A Place for Philosophy? The Rise of Modeling in Statistics. P. Diaconis.Quar. Jour. Appl. Math., 56(4):797-805.

  9. From Shuffling Cards to Walking Around the Building. An Introduction to Markov Chain Theory, Proc. Int. Congress, Berlin, Volume I, Plenary Lectures, 187-204. [PDF]

  10. Magic. In Routledge Encyclopedia of Philosophy

  11. Matchings and Phylogenetic Trees. P. Diaconis & S. Holmes. Proc. Nat. Acad. Sci. of the U.S.A., 95(25):14600-14602. [PDF]



  1. Are There Still Things to Do in Bayesian Statistics? P. Diaconis & S. Holmes, Erkenntnis: Probability, Dynamics and Causality, 45(2-3):145-58.

  2. A Non-Measurable Tail Set. P. Diaconis & D. Blackwell. In Statistics, Probability and Game Theory, Papers in Honor of David Blackwell. T. Ferguson et al. (eds.). IMS, Hayward, pp. 1-5.

  3. Consistency of Bayes Estimates for Non-Parametric Regression: A Review. P. Diaconis & D. Freedman. In D. Pollard, et al. (eds.), Festschrift for Lucien LeCam, pp. 157-66, Springer, New York



  1. Random Walks on Finite Groups: A Survey of Analytic Techniques. P. Diaconis L. Saloff-Coste, Prob. Meas. on Groups XI, H. Heyer (ed.), World Scientific Singapore, 11 pp. 44-75.

  2. The Cutoff Phenomenon in Finite Markov Chains, Proc. Nat. Acad. Sci. of U.S.A., 93(4):1659-1664. [PDF]

  3. Primitive Partition Identities. P. Diaconis, R.L. Graham & B. Sturmfels, Combinatorics - Paul Erdös is Eighty, Bolyai Society Mathematical Studies, Budapest, 2:173-192. [PDF]

  4. Nash Inequaltites for Finite Markov Chains. P. Diaconis & L. Saloff-Coste, Journal of Theoretical Probability, 9:459-510. [PS]

  5. Walks on Generating Sets of Abelian Groups. P. Diaconis & L. Saloff-Coste, Prob. Theory Related Fields, 105:393-421. [PDF]

  6. Logarithmic Sobolev Inequalities for Finite Markov Chains. P. Diaconis & L. Saloff-Coste, Ann. Appl. Prob, 6:695-750. [PDF]

  7. Metrics on Compositions and Coincidences Among Renewal Sequences. P. Diaconis & S. Holmes, S. Janson, S.P. Lalley & R. Pemantle. In D. Aldous, R. Pemantle (eds.), Random Discrete Structures, IMA publications, Springer Verlag, New York, pp. 81-102

  8. Some New Tools for Dirichlet Priors, Proceedings of the Fifth Valencia International Meeting, June 5-9, 1994. J. Bernardo, J. Berger, A. Dawid and F. Smith (eds.) Bayesian Statistics 5 , Oxford University Press,1996, pp. 97-106
    [PDF] OUP link



  1. An Application of Harnack Inequalities to Random Walk on Nilpotent Quotients. P. Diaconis & L. Saloff-Coste, Journal Fourier Analysis and Applications, Kahane Special Issue, 189-207. [PDF]

  2. Riffle Shuffles, Cycles and Descents. P. Diaconis & M. McGrath & J. Pitman, Combinatorica, 15(1):11-29. [PDF]

  3. What Do We Know About the Metropolis Algorithm?. P. Diaconis & L. Saloff-Coste, Proc. 27 Symp. Theory Comp., 112-129. [PDF]

  4. Hammersley's Interacting Particle Process and Longest Increasing Subsequences. P. Diaconis & D. Aldous. Prob. Theory Related Fields, 103:199-213. Abstract [PDF] [PostScript]

  5. NonParametric Binary Regression with Random Covariates. P. Diaconis & D. Freedman. Prob. and Math. Stat., 15:243-273. [PDF]



  1. Spectral Analysis for Discrete Longitudinal Data. P. Diaconis & L. Beckett, Advances in Appl. Math, 103:107-128. [PDF]

  2. On the Eigenvalues of Random Matrices. P. Diaconis & M. Shahshahani, Jour. Appl. Prob, Special 31A:49-62. [PDF]

  3. Gray Codes for Randomization Procedures. P. Diaconis & S. Holmes, Statistics and Computing, 4:287-302. [PDF]

  4. Three Examples of the Markov Chain Monte Carlo Method. P. Diaconis & S. Holmes, Discrete Probability and Algorithms, D. Aldous et al (eds.). 43-56. Springer-Verlag, New York. [PDF]

  5. Rectangular Arrays With Fixed Margins. P. Diaconis & A. Gangolli, Discrete Probability and Algorithms, D. Aldous et al (eds.). 15-42, Springer-Verlag, New York, Abstract

  6. Moderate Growth and Random Walk on Finite Groups, P. Diaconis & L. Saloff-Coste, Geom. Func. Anal., 4(1):1-36.



  1. Non-Parametric Binary Bayesian Regression: A Bayesian Approach. P. Diaconis D. Freedman, Ann. Stat., 21:2108-2137. [PDF]

  2. Forward to the Book: Probability Models and Statistical Analyses For Ranking Data, M. Fligner, J. Verducci ed. , xvii-xxiii, Springer Lecture Notes in Statistics, 80 Springer, New York

  3. Comparison Techniques for Random Walk on Finite Groups. P. Diaconis L. Saloff-Coste, Ann. Prob., 21 (4):2131-2156. [PDF]

  4. Comparison Theorems for Reversible Markov Chains. P. Diaconis L. Saloff-Coste, Ann. Appl. Prob, 3(3):696-730. [PDF]

  5. Efficient Computation of Isotypic Projections for the Symmetric Group, Groups and computation (New Brunswick, NJ, 1991) . P. Diaconis D. Rockmore, DIMACS Series in Disc. Math. and Theor. Comp. Sci., 11:87-104.
    [PDF] [PostScript]



  1. Trailing the Dovetail Shuffle to its Lair. P. Diaconis D. Bayer, Ann. Appl. Prob., 2(2):294-313. [PDF]

  2. Sufficiency as Statistical Symmetry, Proc. 100th Anniversary Americal Mathematical Society, F. Browder (ed.) Mathematics into the Twenty First Century, pg. 15-26, Amer. Math. Soc., Providence

  3. Non Parametric Binary Bayesian Regression. P. Diaconis D. Freedman, Festschrift for Raj Bahaduhr, Indian Statistical Institute

  4. Universal Cycles for Combinatorial Structures. P. Diaconis F. Chung, R.L. Graham, Discrete Math, 110(1-3):43-59. [PDF]

  5. Finite de Finetti Theorems in Linear Modules and Multivariate Analysis, P. Diaconis M.L. Eaton & S. Lauritzan, Scand. Jour. Statist., 19:289-315

  6. An Affine Walk on the Hypercube. P. Diaconis R.L. Graham, Quat. Jour. Analysis, 41(1-2):215-235. [PDF]

  7. Binomial Coefficient Codes Over GF(2). P. Diaconis R.L. Graham, Discrete Math., 106-107:181-188. [PDF]

  8. Analysis of Top to Random Shuffles. P. Diaconis J. Fill & J. Pitman. Combinatorics, Probability Computing, 1:135-155. [PDF]

  9. Eigen-Analysis for Some Examples of the Metropolis Algorithm, P. Diaconis P. Hanlon, Contemporary Math., 138:99-117. [PDF]



  1. Geometric Bounds for Eigenvalues of Markov Chains. P. Diaconis D. Stroock, Ann. Appl. Prob., 1(1):36-61. [PDF]

  2. A Growth Model, A Game, An Algebra, Lagrange Inversion and Characteristic Classes. P. Diaconis W. Fulton. Recondita Math., 49:95-119. [PDF]

  3. Closed Form Summation for Classical Distributions: Variations on a Theme of Demoivre. P. Diaconis S. Zabell, Statistical Sci., 61(3):284-302. [PDF]



  1. Bounds for Tail Probabilities of Weighted Sums of Independent Gamma Random Variables. P. Diaconis M. Perlman, In Topics in Statistical Dependence, H.W. Block et al (eds.), Institute of Mathematical Statistics, Hayward Ca., 147-166

  2. Asymptotic Analysis of a Random Walk on a Hypercube with Many Dimensions. P. Diaconis, R.L. Graham & J. A. Morrison, Random Structures and Algorithms, 1:51-72. [PDF]

  3. Patterned Matrices. Matrix Theory and Applications: Proc. Sympos. Appl. Math., Amer. Math. Soc., 40:37-58

  4. On the Uniform Consistency of Bayes Estimates for Multinomial Probabilities, P. Diaconis & D. Freedman, Ann. Stat., 18(3):1317-1327. [PDF]

  5. Efficient Computation of the Fourier Transform on Finite Groups. P. Diaconis & D. Rockmore, Journ. Amer. Math. Soc., 3(2):297-332.
    AMS link

  6. Examples for the Theory of Strong Stationary Duality with Countable State Spaces. P. Diaconis & J. Fill, Prob. Engin. Info. Sci., 4:157-180.

  7. Strong Stationary Times Via a New Form of Duality. P. Diaconis & J. Fill, Ann. Prob., 18(4):1483-1522. [PDF]

  8. Applications of Groups Representations to Statistical Problems, Proc. International Congress of Mathematician, Kyoto, II 1037-1048, Springer, TOKYO

  9. Finite Fourier Methods: Access to Tools, In Probabalistic combinatorics, Proc. Symposia Appl. Math., B. Bollabos (ed), 44 Amer. Math. Soc. Providence, R.I., 171-194.

  10. Cauchy's Equation and de Finetti's Theorem. P. Diaconis & D. Freedman, Jour. Statist., 17(3):235-250



  1. A Generalization of Spectral Analysis with Application to Ranked Data, Ann. Stat., 17(3):949-979. [PDF]

  2. Applications of Murphy's Elements. P. Diaconis & C. Greene. (1989). Stanford Technical Report, No.335. [PDF]

  3. Fair Dice. P. Diaconis & J. Keller, Amer. Math. Mo., 96:337-339. [PDF]

  4. Methods for Studying Coincidences. P. Diaconis & F. Mosteller, Jour. Amer. Statist. Ann., 84(408):853-861. [PDF]

  5. Bounds for Tail Probabilities of Weighted Sums of Indepedent Gamma Random Variables. In Symposium on Dependence in Statistics and Probability. IMS Lecture Notes Monogr. Ser. 16, Hayward,147-166.



  1. Bayesian Numerical Analysis, Statistical Decision Theory and Related Topics IV , J. Berger, S. Gupta (eds.), 1:163-175, Springer-Verlag, New York. [PDF]

  2. Application of the Method of Moments in Probability and Statistics, In Moments in Mathematics Proc. Symp Appl. Math., Amer. Math. Soc., 37:125-142

  3. Group Representations in Probability and Statistics. P. Diaconis & S. Gupta (ed.). IMS Lecture Notes - Monograph Series, 11 Institute of Mathematical Statistics, Hayward Ca.
    Project Euclid link

  4. Conditional Limit Theorems for Exponential Families with Uniform Asymptotic Estimates and applications to de Finetti's Theorem. P. Diaconis & D. Freedman, J. Theoretical Prob., 1:381-410.

  5. Honest Bernoulli Excursions. P. Diaconis & L. Smith, Journ. Applied Prob., 25(3):464-477. [PDF]

  6. On Merging of Probabilities. P. Diaconis, A. D'Aristotile & D. Freedman, Sankhya, Series A, 50(3):363-380. [PDF]

  7. Recent Progress in de Finetti's Notions of Exchangeability, J. Bernardo, et al., eds., Bayesian Statistics 3: Proceedings of the Third Valencia International Meeting, June 1-5, 1987, pp 111-125.Oxford Science Publications. Oxford: Clarendon Press; New York: Oxford University Press. [PDF]



  1. Time to Reach Stationarity in the Bernoulli-Laplace Diffusion Model, P. Diaconis & M. Shahshahani, SIAM J. Math'l Analysis, 18(1):208-218. [PDF]

  2. Probabilistic-Geometric Theorems Arising from the Analysis of Contingency Tables. P. Diaconis & B. Efron, Contributions to the Theory and Application of Statistics, A Volume in Honor of Herbert Solomon, Academic Press, 103-125. [PDF]

  3. Fred Mosteller as a Mathematical Statistician. P. Diaconis & E. Lehmann, A Statistical Model, S. Fienberg, D. Hoaglin, W. Kruskal, J. Tanur, (eds.), pp 59-80 Springer-Verlag, New York

  4. Projection Pursuit for Discrete Data, Scand. J. Stat.,

  5. Random Walks Arising in Random Number Generation, P. Diaconis, F. Chung & R. L. Graham, Ann. Prob., 15(3):1148-1165. [PDF]

  6. Strong Uniform Times and Finite Random Walks. P. Diaconis & D. Aldous, Adv. in Appl. Math., 8(1):69-97.

  7. The Subgroup Algorithm for Generating Uniform Random Variables, P. Diaconis & M. Shahshahani, Prob. in Eng. and Info. Sci., 1:15-32 [PDF]

  8. A Dozen de Finetti-style Results in Search of a Theory. P. Diaconis & D. Freedman, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, 23(Sup.2):397-423,

  9. Inequalities for Linear Combinations of Gamma Random Variables. P. Diaconis, M. E. Bock & F. Huffer, Canadian J. Stat., 15:387-395.



  1. On Inconsistent Bayes Estimates of Location. P. Diaconis & D. Freedman, Ann. Stat., 14(1):68-87. [PDF]

  2. Products of Random Matrices and Computer Image Generation. P. Diaconis & M. Shahshahani, Contemporary Math., 50:173-182

  3. Products of Random Matrices as They Arise in the Study of Random Walks on Groups. P. Diaconis & M. Shahshahani, Contemporary Math., 50:183-195

  4. On the Consistency of Bayes Estimates. P. Diaconis & D. Freedman, Ann. Stat., 14(1):1-26. [PDF]

  5. An Elementary Proof of Stirling's Formula. P. Diaconis & D. Freedman, Amer. Math'l Monthly, 93:123-125.

  6. Shuffling Cards and Stopping Times. P. Diaconis & D. Aldous, Amer. Math'l Monthly, 93(5):333-348. [PDF]

  7. Applications of Nonommutative Fourier Analysis to Probability Problems. P. Diaconis, et al. Ecolé d' Été de Probabilites de St. Flours, XV-XVII, Springer Lecture Notes in Mathematics, 1362:51-100. Springer-Verlag, Berlin

  8. A Subjective Guide to Objective Chance, P. Diaconis & E. Engel, Statistical Science, 1:171-174.

  9. On Square Roots of the Uniform Distribution on Compact Groups, P. Diaconis &M. Shahshahani, Proc. Amer. Math'l Society, 98:341-348. [PDF]



  1. Quantifying Prior Opinion. P. Diaconis & D. Ylvisaker. Bayesian Statistics 2. Proc. 2nd Valencia Int'l Meeting, 9-83. J. M. Bernardo, M. H. Degroot, D. V. Lindley, A. F. M. Smith (eds.) North-Holland, Amsterdam 133-156.

  2. Some Alternatives to Bayes' Rule. P. Diaconis & S. Zabell, In Information and Group Decision Making, Proc. Second Univ. of Calif. Irvine Conf. Political Economy, B. Grofman, G. Owen (eds.) Jai Press, Greenwich, CT, 25-38. [PDF]

  3. Theories of Data Analysis: From Magical Thinking Through Classical Statistics, Exploring Data Tables, Trends and Shapes, D. Hoaglin, F. Mosteller, J. Tukey (eds.) Wiley, New York, 1-36. [PDF]

  4. Bayesian Statistics as Honest Work, Proc. Berkeley Conf. in Honor of Jerzey Neyman and Jack Kiefer, Volume I. L. LeCam, R. Olshen (eds.) Wadsworth, 53-64.

  5. Testing for Independence in a Two-Way Table: New Interpretations of the Chi-Square Statistic. P. Diaconis & B. Efron, Ann. Stat., 13(3):845-913. [PDF]

  6. The Radon Transform on Z^k_2 . P. Diaconis & R. L. Graham, Pacific J. Math., 118:323-345. [PDF]



  1. On Nonlinear Functions of Linear Combinations. P. Diaconis & M.Shahshahani, SIAM J. Scientific. Comput., 5(1):175-191. [PDF]

  2. Asymptotics of Graphical Projection Pursuit. P. Diaconis & D. Freedman, Ann. Stat., 12(3):793-815. [PDF]



  1. The Mathematics of Perfect Shuffles. P. Diaconis, R. L. Graham & W. M. Kantor, Adv. in Appl. Math., 4(2):175-196. [PDF]

  2. M and N plots. P. Diaconis & J. H. Friedman, Recent Advances in Statistics , H. Rizvi, J. Rustagi, D. Siegmund (eds.), Academic Press, New York, 425-447 [PDF]

  3. Computer Intensive Methods in Statistics. P. Diaconis & B. Efron, Scientific American, 248:116-130.

  4. On Inconsistent Bayes Estimates in the Discrete Case. P. Diaconis & D. Freedman, Ann. Stat., 11(4):1109-1118. [PDF]

  5. Frequency Properties of Bayes Rules. P. Diaconis & D. Freedman, In Scientific Inference, Data Analysis, and Robustness, G. Box, T. Leonard, C. F. Wu (eds.). Academic Press, New York, 105-115.



  1. On the Maximum Difference Between the Empirical and Expected Histograms for Sums. P. Diaconis D. Freedman, Pacific J. Math., 100(2):287-327. [PDF]

  2. On the Difference Between the Empirical Histogram and the Normal Curve for Sums, Part II. P. Diaconis & D. Freedman, Pacific J. Math., 100(2):359-371 (1982). [PDF]

  3. On the Mode of an Empirical Histogram for Sums. P. Diaconis & D. Freedman, Pacific J. Math., 100(2):373-385. [PDF]

  4. de Finetti's Theorem for Symmetric Location Families, P. Diaconis & D. Freedman, Ann. Stat., 10(1):84-189. [PDF]

  5. On Inconsistent M -Estimator. P. Diaconis & D. Freedman, Ann. Stat., 10(2):454-461. [PDF]

  6. Bayes Rules for Location Problems. P. Diaconis & D. Freedman, Statistical Decision Theory and Related Topics III, S. Gupta, J. Berger (ed.), 315-327

  7. Variables on Scatterplots Look More Highly Correlated When the Scales are Increased. P. Diaconis, W. S. Cleveland & R. McGill, Science, 216(4550):1138-1141

  8. Updating Subjective Probability. P. Diaconis & S. Zabell, JASA, 77(380):822-830. [PDF]



  1. Partial Exchangeability and Sufficiency. P. Diaconis & D. Freedman, Proc. Indian Stat. Inst. Golden Jubilee Int'l Conf. Stat.: Applications and New Directions , J. K. Ghosh & J. Roy (eds.), Indian Statistical Institute, Calcutta, 205-236. [PDF]

  2. The Analysis of Sequential Experiments with Feedback to Subjects. P. Diaconis & R.L. Graham, Ann. Stat., 9(1):3-23. [PDF]

  3. Magical Thinking in the Analysis of Scientific Data, Ann. New York Academy of Sci., 364:236-244

  4. On the Permanents of Complements of the Direct Sum of Identity Matrices. P. Diaconis, F. Chung, R. L. Graham & C. L. Mallows. Advances in Applied Math., 2:121-137. [PDF]

  5. On the Statistics of Vision: The Julesz Conjecture. P. Diaconis & D. Freedman, J. Math'l Psychology, 24(2):112-138. [PDF]

  6. Generating a Random Permutation with Random Transpositions, P. Diaconis & M. Shahshahani, Z. Wahr. verw. Gebiete, 57(2):159-179

  7. On the Histogram as a Density Estimator: L_2 Theory, P. Diaconis & D. Freedman, Z. Wahr. verw. Gebiete, 57:453-476

  8. On the Maximum Deviation Between the Histogram and the Underlying Density. P. Diaconis & D. Freedman, Z. Wahr. verw. Gebiete, 58:139-167

  9. The Persistence of Cognitive Illusions: A Rejoinder to L. J. Cohen. P. Diaconis & D. Freedman, Behavioral and Brain Sci., 4:333-334

  10. How Fast is the Fastest Fourier Transform? In Computer Science and Statistics: Proceedings of the 13th Symposium on the Interface, W.F. Eddy (ed.), New York: Springer-Verlag, pp. 43-4



  1. Finite Exchangeable Sequences. P. Diaconis & D. Freedman, Ann. Prob., 8(4):745-764. [PDF]

  2. de Finetti's Theorem for Markov Chains. P. Diaconis & D. Freedman, Ann. Prob., 8:115-130. [PDF]

  3. Average Running Time of the Fast Fourier Transform, Jour. of Algorithms, 1(2):187-208.

  4. de Finetti's Generalizations of Exchangeability. P. Diaconis & D. Freedman, Studies in Inductive Logic and Probability, Volume II, (R. Jeffrey, ed.).



  1. Conjugate Priors for Exponential Families, P. Diaconis & D. Ylvisaker, (1979).
    Ann. Stat., 7(2):269-281. [PDF]

  2. On Rounding Percentages. P. Diaconis & D. Freedman. JASA, 74(366):359-364. [PDF]



  1. Some Tauberian Theorems Related to Coin Tossing. P. Diaconis & C. Stein. Ann. Prob.,6(3):483-90. [PDF]

  2. Statistical Problems in ESP Research. P. Diaconis, C.T. Tart, H.E. Puthoff, & R. Targ. Science Magazine, 202(4373):1145-6. [PDF].



  1. Second-order Terms for the Variances and Covariances of the Number of Prime Factors - Including the Square Free Case. P. Diaconis, F. Mosteller & H. Onishi. Journ. Number Theory , 9(2):187-202. [PDF]

  2. Finite Forms of de Finetti's Theorem on Exchangeability, Synthese, 36(2):271-81.

  3. The Distribution of Leading Digits and Uniform Distribution Mod 1. Ann. Prob., 5(1):72-81. [PDF]

  4. Examples for the Theory of Infinite Iteration of Summability Methods. Canadian Journ. Math., 29:489-97.

  5. Spearman's Footrule as a Measure of Disarray. P. Diaconis & R.L. Graham. Journ. Royal Stat'l Soc. Series B (Methodological), 39(2):262-8. [PDF]



  1. Buffon's Problem with a Long Needle. J. Applied Prob., 13(3):614-618. [PDF]

  2. Protocol issues in Randomized Clinical Trials of Surgical Treatment of Duodenal Ulcer. Costs, Risks and Benefits of Surgery (Barnes, Bunker, Mosteller, eds.), Oxford University Press. Joint with faculty Group on ulcer surgery.





  1. PhD Dissertation: Weak and Strong Averages in Probability and the Theory of Numbers, Harvard University, Cambridge, Massachusetts. May, 1974. [PDF]