Abstract
Let \({\mathop \phi \limits^ \wedge _1}, \ldots ,{\mathop \phi \limits^ \wedge _n}\) be the unnormalized conditional estimates of Φ 1(X(t)),…,Φ n(X(t)) in the cubic sensor problem. For arbitrary n and linearly independent Φ 1,…, Φ n we show that the random variable \(\Phi = \left( {{{\mathop \phi \limits^ \wedge }_1}, \ldots ,{{\mathop \phi \limits^ \wedge }_n}} \right)\) admits a density. This strongly precludes the existence of a finite dimensionally computable realization of the unnormalized conditional density, and it provides an example for a general theorem given in [6].
Mathematics Department, Rutgers University, New Brunswick, NJ 08903; Supported in part by NSF Grant No. MCS-8301880.
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Ocone, D. (1988). Existence of Densities for Statistics in the Cubic Sensor Problem. In: Fleming, W., Lions, PL. (eds) Stochastic Differential Systems, Stochastic Control Theory and Applications. The IMA Volumes in Mathematics and Its Applications, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8762-6_25
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DOI: https://doi.org/10.1007/978-1-4613-8762-6_25
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