X-12-ARIMA

On Joint Fourier-Laplace Transforms

Patrick Fitzsimmons ⁽¹⁾ and Tucker S. McElroy⁽²⁾

ABSTRACT:

Weak convergence of random variables is characterized by pointwise convergence of the Fourier transform of the respective distributions, and in some cases can also be characterized through the Laplace transform. For some distributions, the Laplace transform is easier to compute and provides an alternative approach to the method of characteristic functions that facilitates proving weak convergence. We show that for a bivariate distribution, a joint Fourier-Laplace transform always characterizes the distribution when the second variate is positive almost surely.

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⁽¹⁾ Patrick Fitzsimmons is currently at the University of California, San Diego

⁽²⁾ Tucker S. McElroy is Mathematical Statistican, U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email : Tucker.S.McElroy@census.gov