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The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a given value is as close as possible to its true value.
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Generating Strictly Controlled Stimuli for Figure Recognition Experiments
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Lacunary Series and Strong Approximation.

István Berkes1

  • 1Rényi Mathematical Institute, Reáltanoda u. 13-15, 1053 Budapest, Hungary.

Entropy (Basel, Switzerland)
|February 26, 2025
PubMed
Summary

This study applies Strassen

Area of Science:

  • Probability Theory
  • Statistics

Background:

  • Strong approximation is a key method for proving limit theorems.
  • Lacunary series and conditionally independent sequences are important in probability theory.

Purpose of the Study:

  • To extend the application of strong approximation.
  • To prove limit theorems for lacunary series with specific dependencies.

Main Methods:

  • Utilizing strong approximation techniques.
  • Applying methods to lacunary series with conditionally independent sequences.

Main Results:

  • Established uniform limit theorems for lacunary series.
  • Proved permutation-invariant limit theorems for these series.
Keywords:
conditional independenceexchangeabilitylacunary serieslimit theoremssubsequence principle

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Conclusions:

  • Strong approximation is effective for analyzing complex series.
  • The findings contribute to the understanding of limit theorems in dependent settings.