Information bounds and quickest change detection in decentralized decision systems

The quickest change detection problem is studied in decentralized decision systems, where a set of sensors receive independent observations and send summary messages to the fusion center, which makes a final decision. In the system where the sensors do not have access to their past observations, the previously conjectured asymptotic optimality of a procedure with a monotone likelihood ratio quantizer (MLRQ) is proved. In the case of additive Gaussian sensor noise, if the signal-to-noise ratios (SNR) at some sensors are sufficiently high, this procedure can perform as well as the optimal centralized procedure that has access to all the sensor observations. Even if all SNRs are low, its detection delay will be at most pi/2-1 approximate to 57% larger than that of the optimal centralized procedure. Next, in the system where the sensors have full access to their past observations, the first asymptotically optimal procedure in the literature is developed. Surprisingly, the procedure has the same asymptotic performance as the optimal centralized procedure, although it may perform poorly in some practical situations because of slow asymptotic convergence. Finally, it is shown that neither past message information nor the feedback from the fusion center improves the asymptotic performance in the simplest model

Two Definite Integrals Involving Products of Four Legendre Functions

The definite integrals $\int_{-1}^1x[P_\nu(x)]^4\,\mathrm{d} x$ and $\int_{0}^1x[P_\nu(x)]^2\{[P_\nu(x)]^2-[P_\nu(-x)]^2\}\,\mathrm{d} x$ are evaluated in closed form, where $P_\nu$ stands for the Legendre function of degree $\nu\in\mathbb C$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.Comment: 12 pages, accepted version. Proof of two integral formulae stated in arXiv:1301.1735v4 and arXiv:1506.00318v