Michael Grabchak, Professor

Research Interests

  • Tempered Stable Distributions
  • Infinitely Divisible Processes
  • Heavy Tails
  • Quantitative Finance
  • Probability and Statistics on Alphabets
  • Estimation of Entropy and Related Information Theoretic Quantities

See my page on Google Scholar.

Scientific Software

  1. M. Grabchak and L. Cao (2023). SubTS: Positive Tempered Stable Distributions and Related Subordinators. Ver. 1.0, R Package.
  2. M. Grabchak and L. Cao (2017). SymTS: Symmetric Tempered Stable Distributions. Ver. 1.0, R Package. (Update: Version 1.0-2, 2023)
  3. L. Cao and M. Grabchak (2015). EntropyEstimation: Estimation of Entropy and Related Quantities. Ver. 1.2, R Package.


  1. M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham, Switzerland.

Edited Collections

  1. M. Grabchak, editor (2023). 50 Years of Journal of Applied Statistics: Extreme Value Theory and its Applications in Finance. Taylor and Francis, 15 articles.


  1. M. Grabchak and X. Zhang (2024). On the small jumps of Levy processes and the multivariate Dickman distribution. Preprint.

Published Articles

  1. Y. Xia and M. Grabchak (2024). Pricing multi-asset options with tempered stable distributions. Financial Innovation. To appear.
  2. M. Grabchak and X. Zang (2024). Representation and simulation of multivariate Dickman distributions and Vervaat perpetuities. Statistics and Computing, 34(1): Article 28. Free Version. Preprint.
  3. J. Chang and M. Grabchak (2023). Necessary and sufficient conditions for the asymptotic normality of higher order Turing estimators. Bernoulli, 29(4):3369-3395.
  4. M. Grabchak (2023). How do we perform a paired t-test when we don't know how to pair? The American Statistician, 77(2):127-133.
  5. M. Grabchak and P. Sabino (2023). Efficient simulation of p-tempered alpha-stable OU processes. Statistics and Computing, 33(1): Article 4. Free Version. Preprint.
  6. E. Christou and M. Grabchak (2022). Risk estimation with composite quantile regression. Econometrics and Statistics, DOI 10.1016/j.ecosta.2022.04.004.
  7. M. Grabchak and I.M. Sonin (2022). A zero-one law for Markov chains. Stochastics, 94(5):680-697. Preprint.
  8. Y. Xia and M. Grabchak (2022). Estimation and simulation for multivariate tempered stable distributions. Journal of Statistical Computation and Simulation, 92(3):451-475.
  9. M. Grabchak (2022). Discrete tempered stable distributions. Methodology and Computing in Applied Probability, 24(3):1877-1890. Free Version.
  10. E. Christou and M. Grabchak (2022). Estimation of expected shortfall using quantile regression: A comparison study. Computational Economics, 60(2):725-753. Free Version.
  11. M. Grabchak, S.A. Molchanov, and V. Panov (2022). Around the infinite divisibility of the Dickman distribution and related topics. Zapiski Nauchnykh Seminarov POMI, 115:91-120. Alternate site.
  12. M. Grabchak and E. Christou (2021). A note on calculating expected shortfall for discrete time stochastic volatility models. Financial Innovation, 7: Article 43.
  13. M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015. Preprint.
  14. M. Grabchak (2021). On the transition laws of p-tempered alpha-stable OU-processes. Computational Statistics, 36(2):1415-1436. Preprint. Free Version.
  15. A. Mahzarnia, M. Grabchak, and J. Jiang (2021). Estimation of the minimum probability of a multinomial distribution. Journal of Statistical Theory and Practice, 15(2): Article 24. Free Version.
  16. M. Grabchak (2020). On the simulation of general tempered stable Ornstein-Uhlenbeck processes. Journal of Statistical Computation and Simulation, 90(6):1057-1081. Preprint.
  17. M. Grabchak, M. Kelbert, and Q. Paris (2020). On the occupancy problem for a regime switching model. Journal of Applied Probability, 57(1):53-77. Preprint.
  18. M. Grabchak (2019). Rejection sampling for tempered Levy processes. Statistics and Computing, 29(3):549-558. Preprint, Free Version.
  19. E. Christou and M. Grabchak (2019). Estimation of value-at-risk using single index quantile regression. Journal of Applied Statistics, 46(13):2418-2433.
  20. M. Grabchak and S.A. Molchanov (2019). Limit Theorems for Random Exponentials: The Bounded Support Case. Theory of Probability and Its Applications, 63(4):634-647. (Also printed in Teoriya Veroyatnostei i ee Primeneniya 63(4):779-794. It can be found here.)
  21. M. Grabchak and S.A. Molchanov (2019). The alloy model: Phase transitions and diagrams for a random energy model with mixtures. Markov Processes and Related Fields, 25(4):591-613. Preprint.
  22. G. Decrouez, M. Grabchak, and Q. Paris (2018). Finite Sample Properties of the Mean Occupancy Counts and Probabilities. Bernoulli, 24(3):1910-1941. Preprint.
  23. M. Grabchak (2018). Domains of Attraction for Positive and Discrete Tempered Stable Distributions. Journal of Applied Probability, 55(1):30-42. Preprint.
  24. M. Grabchak and Z. Zhang (2018). Asymptotic Normality for Plug-in Estimators of Diversity Indices on Countable Alphabet. Journal of Nonparametric Statistics, 30(3):774-795.
  25. C. Chen, M. Grabchak, A. Stewart, J. Zhang, and Z. Zhang (2018). Normal Laws for Two Entropy Estimators on Infinite Alphabets. Entropy, 20(5), 371.
  26. M. Grabchak, L. Cao, and Z. Zhang (2018). Authorship Attribution Using Diversity Profiles. Journal of Quantitative Linguistics, 25(2):142-155.
  27. M. Grabchak (2018). A Random Variable That Does Not Belong to a Domain of Attraction, but its Absolute Value Does. The Mathematical Scientist , 43(1):56-59.
  28. M. Grabchak and Z. Zhang (2017). Asymptotic Properties of Turing's Formula in Relative Error. Machine Learning, 106(11):1771-1785. Preprint.
  29. M. Grabchak, E. Marcon, G. Lang, and Z. Zhang (2017). The Generalized Simpson's Entropy is a Measure of Biodiversity. PLoS ONE, 12(3):e0173305.doi:10.1371/journal.pone.0173305. Preprint.
  30. M. Grabchak (2017). A Simple Condition for the Multivariate CLT and the Attraction to the Gaussian of Levy Processes at Long and Short Times. Communications in Statistics -- Theory and Methods, 46(1):446-456. Preprint.
  31. M. Grabchak and V. Cosme (2017). On The Performance of Turing's Formula: A Simulation Study. Communications in Statistics -- Simulation and Computation, 46(6):4199-4209.
  32. Z. Zhang and M. Grabchak (2016). Entropic Representation and Estimation of Diversity Indices. Journal of Nonparametric Statistics, 28(3):563--575. Preprint.
  33. M. Grabchak (2016). On the Consistency of the MLE for Ornstein-Uhlenbeck and Other Selfdecomposable Processes. Statistical Inference for Stochastic Processes, 19(1):29-50.
  34. M. Grabchak (2015). Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes. Journal of Theoretical Probability, 28(1):184-197. Preprint.
  35. M. Grabchak (2015). Three Upsilon Transforms Related to Tempered Stable Distributions. Electronic Communications in Probability, 20(82):1-10. Preprint.
  36. M. Grabchak and S. Molchanov (2015). Limit Theorems and Phase Transitions for Two Models of Summation of iid Random Variables With a Parameter. Theory of Probability and Its Applications, 59(2):222-243. (Also printed in Teoriya Veroyatnostei i ee Primeneniya 59(2):340-364. It can be found here.)
  37. Z. Zhang and M. Grabchak (2014). Nonparametric Estimation of Kullback-Leibler Divergence. Neural Computation, 26(11), 2570-2593.
  38. M. Grabchak (2014). Does Value-at-Risk Encourage Diversification When Losses Follow Tempered Stable or More General Levy Processes? Annals of Finance, 10(4):553-568.
  39. M. Grabchak, Z. Zhang, and D. T. Zhang (2013). Authorship Attribution Using Entropy. Journal of Quantitative Linguistics, 20(4):301-313.
  40. Z. Zhang and M. Grabchak (2013). Bias Adjustment for a Nonparametric Entropy Estimator. Entropy, 15(6), 1999-2011.
  41. M. Grabchak (2012). On a New Class of Tempered Stable Distributions: Moments and Regular Variation. Journal of Applied Probability, 49(4):1015-1035. Preprint.
  42. M. Grabchak and G. Samorodnitsky (2010). Do Financial Returns Have Finite of Infinite Variance? A Paradox and an Explanation. Quantitative Finance, 10(8):883-893. Preprint.

Conference and Other Publications

  1. L. Cao and M. Grabchak (2023). Experience report on using WeBWorK in teaching discrete mathematics. SIGCSE 2023: Proceedings of the 54th ACM Technical Symposium on Computer Science Education V.1, pg. 861-867.
  2. W. Misenheimer and M. Grabchak (2021). A Simple Method for Predicting Winners of NFL Football Games. The Pi-Mu-Epsilon Journal, 15(4):205-214.
  3. L. Cao and M. Grabchak (2019). Interactive Preparatory Work in a Flipped Programming Course. CompEd '19: Proceedings of the ACM Conference on Global Computing Education, pg. 229-235.
  4. L. Cao and M. Grabchak (2014). Smoothly Truncated Levy Walks: Toward a Realistic Mobility Model. IPCCC '14: Proceedings of the 33rd International Performance Computing and Communications Conference. Preprint.
  5. M. Grabchak and S. Molchanov (2013). Limit Theorems and Phase Transitions for Two Models of Summation of iid Random Variables Depending on Parameters (extended abstract). Doklady Mathematics, 88(1):431-434. (Russian translation available in Doklady Akademii Nauk, 451(4), 374-377. It can be found here.)
  6. M. Grabchak, N. Bhamidipati, R. Bhatt, and D. Garg (2011). Adaptive Policies for Selecting Groupon Style Chunked Reward Ads in a Stochastic Knapsack Framework. WWW '11: Proceedings of the 20th International Conference on World Wide Web, pg. 167-176. Preprint.

Technical Reports

  1. M. Grabchak (2012). Limit Theorems For Sequences of Tempered Stable and Related Distributions.

Special Invited Courses

  1. Advanced topics course entitled "Topics in Applied Probability: Tempered Stable Distributions, Ad Placement, and Turing's Formula." A 15-hour course, October 6-17, 2014, Higher School of Economics in Moscow, Russia. Organized by the Laboratory of Stochastic Analysis and its Applications

Conference Talks

  1. AMS Spring Southeastern Sectional Meeting, Atlanta, GA, USA, March 2023 (invited talk)
  2. SIAM Conference on Financial Mathematics and Engineering 2021, Philadelphia, PA, USA (online due to COVID-19), June 2021 (invited talk)
  3. Laboratory of Stochastic Analysis and its Applications Autumn Meeting--2020, Moscow, Russia (online due to COVID-19), October 2020 (invited talk)
  4. 40th Conference on Stochastic Processes and their Applications, Gothenburg, Sweden, June 2018
  5. Laboratory of Stochastic Analysis and its Applications Winter Meeting--2017, Moscow, Russia, December 2017 (invited talk)
  6. 10th Extreme Value Analysis Conference, Delft, Netherlands, July 2017
  7. Heavy Tails and Long Range Dependence: A Conference in honor of Gennady Samorodnitsky's 60th birthday, Paris, France, June 2017 (invited talk)
  8. International Workshop on Applied Probability (IWAP) 2016, Toronto, Canada, March 2016 (invited talk)
  9. XVII April International Academic Conference on Economic and Social Development, Moscow, Russia, April 2016 (invited talk)
  10. The 18th INFORMS Applied Probability Society Conference, Istanbul, Turkey, July 2015 (invited talk)
  11. AMS Joint Mathematics Meeting, San Antonio, TX, January 2015 (invited talk)
  12. "Stochastics, Statistics, Financial Mathematics," a conference in honor of Professor Albert Shiryaev's 80th anniversary, Moscow, Russia, October 2014 (invited talk)
  13. AMS Southeastern Spring Sectional Meeting, Knoxville, TN, March 2014 (invited talk)
  14. AMS Joint Mathematics Meeting, Baltimore, MD, January 2014 (invited talk)
  15. The 17th INFORMS Applied Probability Society Conference, San Jose, Costa Rica, July 2013 (invited talk)
  16. 8th World Congress on Probability and Statistics, Istanbul, Turkey, July 2012
  17. Risk Analysis in Economics and Finance Conference, CIMAT, University of Guanajuato, February 2011
  18. 10th International Vilnius Conference on Probability Theory and Mathematical Statistics, June 2010 (Invited Talk)

Colloquium Talks

  1. Spring Statistics Seminar Series, Mississippi State University, March 2023
  2. Probability and Stochastic Processes Seminar, University of Tennessee Knoxville, September 2021
  3. United Seminar of the Department of Probability, Lomonosov Moscow State University (online due to COVID-19), April 2021
  4. Biodiversity Informatics Global Webinar, September 2017
  5. Mathematical Methods in Finance Seminar, Instituto Nacional de Matematica Pura e Aplicada (IMPA), August 2015
  6. Probability Seminar, Duke University, March 2013
  7. Virginia Military Institute (VMI), October 2012
  8. Bangalore Probability Seminar, Indian Institute of Science, July 2012
  9. Indian Statistical Institute, Delhi, June 2012
  10. Risk Management Seminar, Belk Business School, UNC Charlotte, February 2012
  11. Departmental Colloquium, Department of Mathematics and Statistics at American University, October 2011
  12. Bangalore Probability Seminar, Indian Institute of Science, July 2010
  13. Department of Biostatistics, University of Buffalo, March 2010
  14. Department of Mathematical Science, University of Copenhagen, May 2009

Talks for Undergraduates

  1. Fayetteville State University, March 2012, Ad.
  2. Induction ceremony for UNCC's branch of Pi-Mu-Epsilon, UNC Charlotte, February 2012