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Limit theorems for weighted sums and stochastic approximation processes.

T L Lai1, H Robbins

  • 1Department of Mathematical Statistics, Columbia University, New York, New York 10027.

Proceedings of the National Academy of Sciences of the United States of America
|March 1, 1978
PubMed
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This study develops a Marcinkiewicz-Zygmund strong law and invariance principles for stochastic approximation processes. These findings are achieved using limit theorems for martingale transforms and weighted sums with minimal error assumptions.

Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Statistical Inference

Background:

  • Stochastic approximation processes are widely used in machine learning and optimization.
  • Understanding the convergence properties of these processes is crucial for their reliable application.
  • Existing theories often require strong assumptions on error terms.

Purpose of the Study:

  • To develop a Marcinkiewicz-Zygmund strong law for stochastic approximation processes.
  • To establish weak and strong invariance principles for these processes.
  • To achieve these results under minimal conditions on the error terms.

Main Methods:

  • Establishing limit theorems for martingale transforms.
  • Utilizing results for weighted sums.

Related Experiment Videos

  • Applying these theorems to analyze stochastic approximation processes.

    • Main Results:

    • A Marcinkiewicz-Zygmund strong law of large numbers is established.
    • Weak and strong invariance principles are proven.
    • The results hold under minimal assumptions on the errors, enhancing their applicability.

    Conclusions:

    • The developed theory provides a robust framework for analyzing stochastic approximation processes.
    • The minimal conditions on errors broaden the scope of applicability for these fundamental results.
    • This work contributes to the theoretical understanding of convergence in stochastic algorithms.