STAT 523: Advanced Theory of Statistical Inference

Spring Quarter 2020

Potential Topics, with some Selected and Random References

The following list is meant to be suggestive, and certainly does not exhaust all possible topics. Within any of these general topics, you will need to narrow the focus of your talk and project. I would be glad to talk with you about your choice of a topic for the project.

1. Martingale Central Limit Theorems
2. Applications of Stochastic Calculus
3. Diffusion processes
4. Variants of Brownian motion: brownian excursion, brownian meander, skew brownian motion.
5. Local time of Brownian motion and other processes
6. Central limit theorems in Banach spaces; multiplier CLT's.
7. Kac's formula; connections between probability and integral and differential equations.
8. Discrete probability and group theory
9. Coupling Methods and Convergence to Poisson
10. Self-Similar processes and fractional Brownian motion
11. Embeddings and strong approximations
12. Limit distribution theory for shape constrained estimation
13. Continuity of Gaussian processes and sample path properties

1. Martingale Central Limit Theorems
   • Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist., 9, 79-94.

   • Rebolledo, R. (1980). Central limit theorems for local martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 51, 269 - 286.

   • Gaenssler, P. and Haeusler, E. (1979). Remarks on the functional central limit theorem for martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 237 - 243.

2. Applications of Stochastic Calculus
   • Duffie, D. (1988). Security Markets - Stochastic Models. Academic Press, San Diego.

   • Samuelson, P. A. (1973). Mathematics of speculative price. SIAM Review 15, 1 - 42.

   • Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer-Verlag, New York.

   • Taylor, H. M. (1975). A stopped brownian motion formula. Ann. Probability 3, 234-246.

   • Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.

   • Andersen, P.K., Borgan, O., Gill, R. D., and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer-Verlag, New York.

3. Diffusion Processes
   • Freedman, D. (1971). Brownian Motion and Diffusion. Holden-Day, San Francisco.

   • Ito, K. and McKean, H. P. (1974). Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.

   • Williams, D. (1979). Diffusions, Markov Processes, and Martingales: I. Wiley, New York.

   • Williams, D. and Rogers, L.C. (1987). Diffusions, Markov Processes, and Martingales: II. Ito Calculus Wiley, New York.

   • Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.

   • Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion. Springer-Verlag.

4. Variants of Brownian motion: brownian excursion, brownian meander, skew brownian motion.
   • Chung, K. L. (1976). Excursions in Brownian motion. Ark. Mat. 14, 155 - 177.

   • Chung, K. L. (1975). Maxima in Brownian excursions. Bull. Amer. Math. Soc. 81, 742-745.

   • Durrett, R. T. and Iglehart, D. L. (1977). Functionals of Brownian meander and excursion. Ann. Probability 5, 130 - 135.

   • Durrett, R. T. and Iglehart, D. L., and Miller, D. R. (1977). Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5, 117-129.

   • Harrison, J.M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Probability 9, 309-313.

   • Iglehart, D. L. (1974). Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2, 608-619.

   • Kaigh, W. D. (1974). A conditional local limit theorem and its application to random walk. Bull. Amer. Math. Soc. 80, 769-770.

   • Kaigh, W. D. (1978). An elementary derivation of the distribution of the maxima of Brownian meander and Brownian excursion. Rocky Mountain J. Math. 8, 641-645.

   • Keilson, J. and Wellner, J. A. (1978). Oscillating Brownian motion. J. Appl. Probability 15, 300-310.

   • Kennedy, D. P. (1976). The distributon of the maximum Brownian excursion. J. Appl. Probability 13, 371-376.

   • Vervaat, W. (1979). A relation between Brownian bridge and brownian excursion. Ann. Probability 7, 143-149.

   • Louchard, G. (1984). Kac's formula, Levy's local time, and Brownian excursions. J. Appl. Prob. 21 21, 479 - 499.

   • McKean, Jr., H.P. (1962/1963). Excursions of a non-singular diffusion. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1, 230-239.

   • Vervaat, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7, 143-149.

   • Koning, A. J. and Protasov, V. (2003). Tail behaviour of Gaussian processes with applications to the Brownian pillow. J. Mult. Anal. 87, 370-397.

5. Local time of Brownian motion and other processes
   • Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probability 8, 1-67.

   • McKean, H.P. (1975). Local time. Adv. in Math. 16, 91-111.

   • McKean, H.P. (1969). Stochastic Integrals. Academic Press.

   • Williams, D. (1976). On a stopped brownian motion formula of H. M. Taylor. Seminaire de Probabilites X: Lecture Notes in Mathematics 511, 235-239. Springer-Verlag.

   • Marcus, M. B. and Rosen, J. (2006). Markov processes, Gaussian processes, and local times. Cambridge University Press, Cambridge.

6. Central limit theorems in Banach spaces; multiplier CLT's.
   • Ledoux, M. and Talagrand, M. (1992). Probability in Banach Spaces. Springer-Verlag.

   • de la Pena, V. H. and Gine, E. (1998). Decoupling: From Dependence to Independence. Springer-Verlag, New York.

   • Ledoux, M. and Talagrand, M. (1988). Un critere sur les petite boles dans le theoreme limite central. Probability Theory and Related Fields 77, 29 - 47.

   • Ledoux, M. and Talagrand, M. (1986). Conditions d'integrabilite pour les multiplicateurs dans le TLC Banachique. Annals of Probability 14, 916-921.

7. Kac's formula; connections between probability and integral and differential equations.
   • Durrett, R. (1984). Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, California.

   • Kac, M. (1951). On some connections between probability theory and differential and integral equations. Proceedings of the Second Berkeley Symposium on Probability Theory and Mathematical Statistics, 189-215. University of California Press, Berkeley.

   • Bass, R. F. (1995). Probabilistic Techniques in Analysis. Springer-Verlag, New York.

8. Discrete probability and group theory
   • Aldous, D. and Diaconis, P. (1995). Hammersley's interacting particle process and longest increasing subsequences. Probability Theory and Related Fields 103, 199-213.

   • Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Advances in Mathematics 26, 206-222.

   • Rains, E. (1998). Increasing subsequences and the classical groups. Electron. J. Combin. 5, paper 12.

   • Fulman, J. (2004). A card shuffling analysis of deformations of the Plancherel measure of the symmetric group. Electronic Journal of Combinatorics 11, #R21

   • Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley's process. Ann. Probab. 34, 1273-1295.

   • Wellner, J. A. (2002). On longest increasing subsequences and random Young tableaux: experimental results and recent theorems. Technical Report, UW Department of Statistics.

9. Coupling Methods and Convergence to Poisson
   • Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford University Press, Oxford.

   • Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer-Verlag.

   • Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.

10. Self-Similar processes and fractional Brownian motion
    • O'Brien, G. L. and Vervaat, W. (1985). Self-similar processes with stationary increments generated by point processes. Ann. Probability 13, 28 - 52.

    • Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes. Z. Wahrsch. verw. Gebiete 50, 5 - 25.

    • Vervaat, W. (1985). Sample path properties of self-similar processes with stationary increments. Ann. Probability 13, 1- 27.

11. Embeddings and strong approximations
    • Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the non-asymptotic viewpoint. Ann. Probability 17, 239 - 256.

    • Csorgo, M., Csorgo, S., Horvath, L., and Mason, D. (1986). Weighted empirical and quantile processes. Ann. Probability 14, 31 - 85.

    • Csorgo, M. and Horvath, L. (1986). Approximations of weighted empirical and quantile processes. Statistics and Probability Letters 4, 275-??.

    • Dudley, R. M. and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrschein. verw. Gebiete 62, 509-522.

    • Mason, D. M. and van Zwet, W. R. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probability 15, 871-884.

    • Mason, D. M. (1998). Notes on the KMT Brownian bridge approximation to the uniform empirical process. Preprint.

    • Mason, D. M. (1998). An exponential inequality for a weighted approximation to the uniform empirical process with applications. Preprint.

12. Limit distribution theory for shape constrained estimation
    • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16, 31-41.

    • Groeneboom, P. (1985). Estimating a monotone density. Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, Vol. II; Berkeley, California, 1983, 539-555.

    • Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81, 79-109.

    • Groeneboom, P. and Jongbloed, G. and Wellner, J. A. (1999). Integrated Brownian motion conditioned to be positive. Ann. Probab. 27, 1283-1303.

    • Groeneboom, P. and Jongbloed, G. and Wellner, J. A. (2001). A canonical process for estimation of convex functions: the ``invelope'' of integrated Brownian motion $+ t^4$. Ann. Statist. 29, 1620-1652.

    • Groeneboom, P. and Wellner, J.A. (2001). Computing Chernoff's distribution. J. Comput. Graph. Statist. 10, 388-400.

13. Continuity of Gaussian processes and sample path properties
    • Adler, R. J. (1990). An Introduction to continuity, extrema, and related topics for general Gaussian processes. IMS Lecture Notes - Monograph Series, 12. IMS, Hayward CA.

    • Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207-216.

    • Dudley, R.M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1, 290-330.

    • Dudley, R.M. (1973). Sample functions of the Gaussian process. Ann. Probab. 1, 66 - 103.

    • Fernique, X. (1997). Fonctions aleatoires gaussiennes, veceurs aleatoires gaussiens. Universite de Montreal Centre de Recherches. Montreal, QC.

    • Sudakov, V. N. and Cirelson, B. S. (1974). Extremal properties of half-spaces for spherically invariant measures. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 41, 14-24.

    • Talagrand, M. (1987). Regularity of Gaussian processes. Acta Math. 159, 99-149.

    • Talagrand, M. (2005). The Generic Chaining. Springer, New York.