Noncentral Chi-Square: Some Observations on Recurrence

Almost sure stability and stabilization of discrete-time stochastic systems

“…5.2]) are obtained by numerical computation and inevitably involve numerical error, which should be kept in mind for applications, though their accuracy is satisfactory in many cases and could be improved. The residual δ(ε β ,ā k , b k , α) in (28) is a complicated function that involves analysis and computation of (expected) logarithm of noncentral χ 2 random variables (see, e.g., [35,36]), for which the upper bound ∆(λ g , ε β , α) is, instead, computed and used in applications. It is also noticed that complex noise, which is often encountered in communications (see, e.g., [37][38][39]), may be used to stabilize an unstable system.…”

“…The almost sure stability problem of system (35) can be then formulated as the generalized eigenvalue minimization problem [30, (1-5), pp. [1][2][3][4][5] min β subject to (36), (37) and (39),…”

Letters to the Editor

“…See also Siegel (1985) and Jones (1987). Indeed, in the same issue of this journal in which Hjort's article appears, Cohen (1988) mentioned &(A) and referred to Siegel's (1979) (1) It contained (a) an existence proof followed by a direct construction of the distribution for Z, , called the purely eccentric distribution, or the noncentral chi squared with zero degrees of freedom; (b) a straight verification of ( I ) , using the end result from (a); and (c) some graphs and complementary remarks. The argument in (b) provides the simplest way to prove (I).…”

Algorithm 939

“…These functions also occur in statistics and probability theory, where they are called noncentral chi-square (semi-integer μ) or noncentral gamma cumulative distributions [Ashour and Abdel-Samad 1990;Cohen 1988;Dyrting 2004;Knüsel and Bablok 1996;Robertson 1976;Ross 1999]. The central gamma cumulative distribution is in fact the incomplete gamma function, and the relation of the Marcum functions to the incomplete gamma functions is given in Section 2.1.…”