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TAUBERIAN THEOREMS FOR MATRIX REGULAR VARIATION.

M M Meerschaert1, H-P Scheffler2

  • 1Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, mcubed@stt.msu.edu , URL : http://www.stt.msu.edu/~mcubed/

Transactions of the American Mathematical Society
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This summary is machine-generated.

This study extends Karamata's Tauberian theorem to matrix-valued functions, linking their asymptotic behavior to their Laplace-Stieltjes transforms. This provides new tools for analyzing matrix functions and their applications in time series analysis.

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Area of Science:

  • Mathematical Analysis
  • Functional Analysis
  • Time Series Analysis

Background:

  • Karamata's Tauberian theorem connects function asymptotics with their Laplace-Stieltjes transforms using regular variation.
  • Existing theorems primarily address scalar-valued functions.

Purpose of the Study:

  • To establish a matrix-valued analogue of Karamata's Tauberian theorem.
  • To explore the asymptotic properties of matrix-valued functions and their transforms.

Main Methods:

  • Generalization of regular variation to matrix-valued functions.
  • Application of Tauberian theorem principles to matrix transforms.

Main Results:

  • A novel Tauberian theorem for matrix-valued functions is established.
  • The relationship between the asymptotic behavior of matrix functions and their Laplace-Stieltjes transforms is clarified.

Conclusions:

  • The developed theorem provides a powerful extension for analyzing matrix-valued functions.
  • Potential applications in advanced time series analysis are highlighted.