Gambling using a finite state machine

(17) holds for any g in C p , and the rightmost two terms go to zero as n +m, it follows from (10) that MVJw,) + A,AT. To show that (9) converges to zero if o is not a signal frequency, it suffices to set A , = 0. Then o1 is no longer a signal frequency. In this case, (17) is bounded below by zero and above by only the rightmost two terms, both of which go the zero as n + W. This proves the theorem. 0 This result justifies the use of the MV(n) spectrum in a wide variety of unknown stationary or nonstationary noise environments. The proof provided explicit error bounds, (171, associated with other signal components as well as the noise. Notice that the p x p matrix (l/n2)E,(ol)*T,,E,(o,) is simply the weighted discrete Fourier transform (DFT) estimate of the signal point spectrum, where the signal covariance data has been weighted by a triangular window. It follows from (17) that the DFT " spectrum " (l/n2)E,(~,~*TnE,(wl) also converges to the signal point spectrum as n +m for a large class of nonsta-tionary as well as stationary noise processes. If TNn corresponds to a nonstationary noise process, this quantity is not a spectrum in the usual sense. In any case, it also provides an upper bound for the MV(n) spectrum for each value of n. REFERENCES [l] V. M. Adamjan, " Asymptotic properties for positive and Toeplitz matrices and operators, " in Operator Theoty y: Aduances and Applications , vol. 43. [2] J. Capon, " High-resolution frequency-wavenumber spectrum analysis , " A geometric approach to the maximum likelihood spectral estimator for sinusoids in noise, " IEEE Trans. A new approach for determining the spectral data of multichannel harmonic signals in noise, " Math. and properties of the Gauss model on the half-line related to them, " Teoret. Abstract-Sequential gambling schemes in which the amount wagered on the future outcome is determined by a finite state (FS) machine are defined and analyzed. It is assumed that the FS machine determines the fraction of the capital wagered at each time instance i on the outcome at the next time instance, i + 1, and that wagers are paid at even odds. The maximal capital achieved by any FS machine is found and its dependence on an empirical entropy measure, H%), defined as the finite state complexity of …