Minimax prediction of random processes with stationary increments from observations with stationary noise

Minimax Interpolation Problem for Random Processes with Stationary Increments

The problem of mean-square optimal estimation of the linear functional AT ξ = ∫ T 0 a(t)ξ(t)dt that depends on the unknown values of a continuous time random process ξ(t), t ∈ R, with stationary nth increments from observations of the process ξ(t) at time points t ∈ R \ [0;T ] is investigated under the condition of spectral certainty as well as under the condition of spectral uncertainty. Formu...

Representations of Gaussian processes with stationary increments

Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

The problem of optimal estimation of the linear functionals Aξ = ∑∞ k=0 a(k)ξ(k) and AN ξ = ∑N k=0 a(k)ξ(k) which depend on the unknown values of a stochastic sequence ξ(m) with stationary nth increments is considered. Estimates are obtained which are based on observations of the sequence ξ(m) + η(m) at points of time m = −1,−2, . . ., where the sequence η(m) is stationary and uncorrelated with...

Memory-Universal Prediction of Stationary Random Processes

We consider the problem of one-step-ahead prediction of a real-valued, stationary, strongly mixing random process fXig1i= 1. The best mean-square predictor of X0 is its conditional mean given the entire infinite past fXig 1 i= 1. Given a sequence of observations X1 X2 XN , we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasi...

Growth Rate of Gaussian Processes with Stationary Increments

Communicated by David Blackwell, November 3, 1969 1. Statement of results. Let (Y, «, t^O) be a real, separable Gaussian process with stationary increments, mean 0, and F0 = 0. Let 2Q(t) be the variance of Yt and define Xt = Yt/(2Q(t)yi\ THEOREM 1. Suppose there exists a nonnegative function v(t) such that Q(s + t) Q(s) (*) lim = 1 uniformly in s t~*oo v(s + 0 "~ () and there exist positive con...