# Accession Number:

## AD0655594

# Title:

## ON THE COEFFICIENTS OF A WIENER CANONICAL EXPANSION FOR THE LIKELIHOOD FUNCTION OF A CONTINUOUS MARTINGALE.

# Descriptive Note:

## Technical rept.,

# Corporate Author:

## INFORMATION RESEARCH ASSOCIATES INC LEXINGTON MASS

# Personal Author(s):

# Report Date:

## 1967-06-12

# Pagination or Media Count:

## 21.0

# Abstract:

Analytical forms for the coefficients and for the sum of the squares of the coefficients of a Wiener Canonical Hermite-Laguerre expansion of the likelihood function of a stochastic process may be readily obtained if that process is a zero initial-value stationary wide-sense, orthogonal-increment continuous Martingale. These coefficients are simple functions of the parameter of the process, sigma subscript o, superscript 2. It is shown that the coefficients containing odd subscripts vanish as expected, while if sigma subscript o, superscript 2 1, all coefficients vanish except the first one i.e., the coefficient for the zeroth order Hermite polynomials. This suggests a matching possibility, whereby the process amplitude variation may be matched to the expansion. The expression for the asymptotic value of the coefficients for large order index is derived and is shown to behave as the inverse fourth root of the index. Author

# Descriptors:

# Subject Categories:

- Statistics and Probability
- Cybernetics