It is shown that the asymptotic behavior of the tail of the sum distribution of a finite number of individually log-normal distributed variates displays the log-normal character of those variates with maximum logarithmic variance. A quantitative definition of the sum-distribution "tail" is established in terms of upper bounds on the relative error or deviation from the asymptotic log-normal property.

The sample covariance matrix arising out of finite memory linear least squares estimation over a set of equally spaced time points, is inverted by spectral methods (operationally referred to as theztransform). It is shown that the complexity of the problem depends only upon the complexity of the input correlation function. The final solution is shown to reduce to the inversion of a triangular system of linear equations of an order less than half the degree of the denominator of the input power spectral density function.

The present study concerns the optimal filtering of a class of input time series in which the amplitude is modulated by uniformly-pulsed periodic functions. A uniform sampling of the output at a period equal to the pulsing period displays the property of time invariance. The consequent usage of bilateral Fourier-Laplace transformations and the separability of terms implicit in the time-invariant nature of the processing effectively inverts the Wiener-Hopf equation and solves the problem. The weighting function is shown to be the sum of the Wiener-Zadeh-Ragazzini solution for the unmodulated case and set of appropriately-weighted delta-function-derivative terms occurring at each end point of the pulsing intervals.