Showing papers by "Robert Liptser published in 1978"
01 Jan 1978
TL;DR: In this article, the problem of filtering, interpolation and extrapolation for conditionally Gaussian random processes (θ ξ) in continuous time t ≥ O is investigated for random sequences with discrete time t = 0, Δ, 2Δ,...., having the property of conditional Gaussianness.
Abstract: The two previous chapters dealt with problems of filtering, interpolation and extrapolation for the conditionally Gaussian random processes (θ ξ), in continuous time t ≥ O. In the present chapter these problems will be investigated for random sequences with discrete time t = 0, Δ, 2Δ,. . ., having the property of “conditional Gaussianness” as well.
12 citations
01 Jan 1978
TL;DR: The present section deals with the problem of linear estimation of unobservable components of a multidimensional stationary wide-sense process (discrete time) with rational spectral density in the components accessible for observation.
Abstract: The objective of this chapter is to show how the equations of optimal nonlinear filtering obtained for conditionally Gaussian random sequences can be applied to solving various problems of mathematical statistics. In particular, the present section deals with the problem of linear estimation of unobservable components of a multidimensional stationary wide-sense process (discrete time) with rational spectral density in the components accessible for observation.
3 citations
01 Jan 1978
TL;DR: In this paper, the authors considered the problem of linear estimation of processes with continuous time and used the concept of a wide-sense Wiener process to obtain the optimal mean square linear estimate.
Abstract: In the previous chapter the interrelation between properties in the ‘wide’ and in the ‘strict’ sense, which is frequently applied in probability theory, was used in finding optimal linear estimates for stationary sequences with rational spectra. Thus it was enough for our purposes to consider the case of Gaussian sequences (Lemma 14.1) for the construction of the optimal mean square linear estimate. This technique will now be used in problems of linear estimation of processes with continuous time. Here the consideration of the concept of a wide-sense Wiener process turns out to be useful.
1 citations
01 Jan 1978
TL;DR: In this paper, the authors considered a random process with unknown parameters, where ξ = (ξι), 0 ≤ t ≤ T ≤ T, and α t = (α;1(t),, α; N (t)) is the known vector function with the measurable deterministic components α; i (t), i = 1, N.
Abstract: Let ξ = (ξι), 0 ≤ t ≤ T, be a random process with
$${\xi _\iota } = \sum\limits_{i = 1}^N {{\alpha _i}\left( t \right)} {\theta _i} + {\eta _\iota }$$
(171)
, where θ = (θ 1,, θ N) is a vector column of the unknown parameters, -∞ < θ i < ∞, i = 1,, N, and α t = (α;1(t),, α; N (t)) is the known vector function with the measurable deterministic components α; i (t), i = 1,, N
01 Jan 1978
TL;DR: In this paper, the authors described observable random processes X = (ξ t ), t ≥ 0, which possessed continuous trajectories and had properties analogous, to a certain extent, to those of a Wiener process.
Abstract: In the previous chapters we described observable random processes X = (ξ t ), t ≥ 0, which possessed continuous trajectories and had properties analogous, to a certain extent, to those of a Wiener process. Chapters 18 and 19 will deal with the case of an observable process that is a point process whose trajectories are pure jump functions (a Poisson process with constant or variable intensity is a typical example).
01 Jan 1978
TL;DR: In this paper, the results obtained in Section 14.3 for linear control problems (using incomplete data) with quadratic performance index are extended to the case of continuous time, and they are shown to be equivalent to the results in Section 7.
Abstract: In this section the results obtained in Section 14.3 for linear control problems (using incomplete data) with quadratic performance index are extended to the case of continuous time.