Research & Publications

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Research Overview

My research lies at the intersection of stochastic processes, fractional calculus, and modern statistical learning. I develop and analyze mathematical models that capture long-range and semi-long-range dependence, with a particular emphasis on tempered fractional processes, random fields, and functional time series. These tools are useful in modeling real-world systems that exhibit memory, heavy tails, or multiscale behavior.

A major component of my current research focuses on stochastic optimization methods, especially stochastic gradient descent (SGD) and its variants, under the influence of structured noise and tempered perturbations. I study how the statistical and dynamical properties of correlated heavy-tailed noise with long memory and tempering affect convergence, stability, and generalization in high-dimensional machine learning algorithms.

My recent work explores the integration of these stochastic models with deep learning architectures, including Transformers and structured state-space models (S4). This research aims to provide a rigorous theoretical framework for understanding optimization and training dynamics in neural networks, while also contributing practical insights for designing more robust learning systems under uncertainty.

Research Interests

  • Stochastic processes with long-range and semi-long-range dependence
  • Tempered fractional processes
  • Functional time series, including dependence structures, forecasting, and spectral analysis
  • Stochastic gradient descent (SGD) and perturbed optimization methods
  • Structured noise and memory effects in high-dimensional learning algorithms
  • Deep learning architectures, including Transformers and structured state-space models (S4)
  • Mathematical foundations of neural networks and their training dynamics under stochastic perturbations
  • Applications in time series analysis, statistical machine learning, and complex data systems

Working Papers   

  1. Fili, M, Sabzikar, F., Hu,G. Tempered Heavy-tailed Correlated Noise in Stochastic Optimization, Submitted to NeurIPS 2025  
    We propose a perturbed gradient method based on tempered fractional stable noise that improves robustness and convergence in nonconvex optimization.
  2. Burnecki, K., Chechkin, A., Sabzikar, F. Tempered Fractional Langevin-Type Models,  Preprint (not yet submitted)  
    This work introduces new Langevin-type models driven by tempered fractional operators and noise, offering analytical insights into nonlocal memory and anomalous diffusion.

Recent Talks

  1. How does tempering affect the local and global properties of fractional Brownian motion? (Talk_ISU_Mathematics_2021_0.pdf)
  2. Stochastic processes with semi-long range dependence (slides)
  3. Nonparametric Regression Under Semi-Long Range Dependence (slides)
  4. What is Fractional Calculus? (slides)

Journal papers and preprints

  1. Macioszek, K., Sabzikar, F. Burnecki, K. (2025): Testing of Tempered Fractional Brownian Motions, BioPhysMath, 1-23. 
    Biswas, E., Sabzikar, F., Phillips, P.C.B. (2024): Boosting the HP filter for trending time series with long-range dependence, Econometric Reviews,1--39.
    Sabzikar, F., Kokoszka, P. (2023). Tempered functional time series, Journal of Time Series Analysis, Vol. 44, 280-293.
  2. Sabzikar, F., Kabala, J., Burnecki, K. (2022). Tempered fractionally integrated process with stable noise as a transient anomalous diffusion model, Journal of Physics A: Mathematical and Theoretical, Vol. 55, 174002. link
  3. Kabalaa, S. J., Burnecki, K., Sabzikar, F. (2021). Tempered linear and non-linear time series models and their application to heavy-tailed solar flare data, Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 31, 113124. link
  4. De Brabanter, K., Sabzikar, F. (2021), Asymptotic theory for regression models with fractional local to unity root errors, Metrika, Vol. 84, 997-1024.link
  5. Boniece, B. C., Didier, G., Sabzikar, F. (2021), Tempered fractional Brownian motion: wavelet estimation, modeling and testing, Applied and Computational Harmonic Analysis, Vol. 51, 461-509. link
  6. Didier, G., Kanamori, S., Sabzikar, F. (2021), On multivariate fractional random fields: tempering and operator-stable laws, Journal of Mathematical Analysis and Applications, Vol. 495(1), 124659. link
  7. Azmoodeh, E., Mishura, Y., Sabzikar, F. (2021), How does tempering affect the local and global properties of fractional Brownian motion?, Journal of Theoretical probability, Vol. 35, 484-527.  link
  8. Anderson, P. Meerschaert, M. M., Sabzikar, F. (2021), Parsimonious time series modeling for high-frequency climate data, Journal of Time Series Analysis, Vol. 42, 442-470. link
  9. Beran, J., Sabzikar, F., Surgailis, D., Telkmann, K. (2020), On the empirical process of tempered moving averages, Statistics and Probability letters, Vol. 167, 1-10. link   
  10. Boniece, B. C., Didier, G., Sabzikar, F. (2020), On fractional Lévy processes: tempering, sample path properties and stochastic integration, Journal of Statistical Physics, Vol. 178, 954–985. link
  11. Sabzikar, F., Phillips, P. C. B., Wang, Q. (2020), Asymptotic theory for near integrated processes driven by tempered linear processes, Journal of Econometrics, Vol. 216, 192–202. link
  12. Sabzikar, F., McLeod, A. I. and Meerschaert, M. M. (2019), Parameter estimation for ARTFIMA time series, Journal of Statistical Planning and Inference, Vol. 200, 129–145. link
  13. Sabzikar, F., Surgailis, D. (2018), Invariance Principles for Tempered Fractionally Integrated Processes, Stochastic Processes and their Applications, Vol. 128, 3419–3438. link
  14. Sabzikar, F., Surgailis, D. (2018), Tempered fractional Brownian and stable motions of second kind, Statistics and Probability Letters, Vol. 132, 17–27. link
  15. Meerschaert, M. M., Sabzikar, F. (2016), Tempered fractional stable motion, Journal of Theoretical Probability, Vol. 29, 681–706. link
  16. Sabzikar F. (2015), Tempered Hermite Process, Modern Stochastics: Theory and Applications, Vol 2, 327–341. link
  17. Sabzikar, F., Meerschaert, M. M., Jinghua Chen, J. (2015), Tempered Fractional Calculus, Journal of Computational Physics, Vol. 293, 14 28, Special Issue on Fractional Partial Differential Equations. link
  18. Meerschaert, M. M., Sabzikar, F., Phanikumar, M. S., Zeleke, A. (2014), Tempered fractional time series model for turbulence in geophysical flows, Journal of Statistical Mechanics: Theory and Experiment, p. P09023. link
  19. Meerschaert, M. M., Sabzikar, F. (2014), Stochastic integration for tempered fractional Brownian motion, Stochastic Processes and their Applications, Vol. 124, 2363–2387. link
  20. Meerschaert, M. M., Sabzikar, F. (2013), Tempered fractional Brownian motion, Statistics and Probability Letters, Vol. 83, 2269–2275. link