An Estimation of Function in Gaussian Stationary Noise: New Spectral Condition
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AN ESTIMATION OF FUNCTION IN GAUSSIAN STATIONARY NOISE: NEW SPECTRAL CONDITION V. N. Solev∗
UDC 519.2
Lower and upper bounds are constructed for the minimax risk in the problem of estimating unknown pseudo-periodic function observed in stationary noise with spectral density satisfying new spectral conditions. Bibliography: 13 titles.
1. Formulation of the problem The present paper continues paper [13]. We keep the notation of the latter with understandable changes and give only brief explanations of the conclusions following directly from the results of that paper. Assume that on the growing segment [−T, T ], a Gaussian process y(t) is observed and dy(t) = s(t) dt + dx(t),
t ∈ [−T, T ].
(1)
Here, s(t) is an unknown function lying in a given convex centrally symmetric subset L∗ of the Banach space L of locally square summable functions s such that x+1 2 |s (t)|2 dt < ∞, (2) sL = sup x
x
and x(t) is a Gaussian process with stationary increments, zero mean, and spectral density f (for details, see [1, 2]). To avoid terminological confusion, we recall that for a (not necessarily stationary) process x with stationary increments, its spectral density f coincides with spectral d x being a generalized process and even a generalized stationary one. density of its derivative dt Moreover, ∞ f (u)) du < ∞. (3) 1 + u2 −∞
Let sT be estimate of the unknown function s constructed from observations (1), sT ∈ L∗ . The risk of using the estimate sT is measured by the quantity sT ; L∗ ) = sup Es,f sT − s2L . RT (
(4)
s∈L∗
Let RT∗ be the minimax risk,
RT∗ (L∗ ) = inf RT ( sT ) . sT
(5)
sT ; L∗ ), The problem is to find the estimate sT of the same order of smallness of the quantity RT( as the minimax risk as T → ∞, sT ) ≤ C RT∗ . RT∗ ≤ RT (
(6)
Throughout the paper, we deal with random variables ∞ ϕ(t) dx(t) x [ϕ] = −∞ ∗
St.Petersburg Department of the Steklov Mathematical Institute; St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 222–232. Original article submitted November 10, 2018. 1072-3374/20/2511-0147 ©2020 Springer Science+Business Media, LLC 147
defined, e.g., for a linear set S of functions ϕ such that 2
ϕ∈L ,
C(ϕ) |ϕ(u)| ≤ , where ϕ(u) = 1 + u2 2
∞
e− iut ϕ(t) dt.
(7)
−∞
Here, 2 E x ψ = 0, E x ψ =
∞
2 f (u) du < ∞, ψ ∈ S. ψ(u)
(8)
−∞
By virtue of the model (1), the random variables ϕ ∈ D(T ) = {ϕ : ϕ ∈ S supp ϕ ⊂ [−T, T ]}
y[ϕ] = s[ϕ] + x[ϕ],
(9)
are available for observation. Here, ∞
∞ ϕ(t) dy(t),
y[ϕ] =
s[ϕ] =
−∞
s(t) ϕ(t) dt, −∞
2. Choice of parametric set L∗ Let Λ be a countable subset of R1 , satisfying the separability condition τ = τ (Λ) = inf |u − v| > 0.
(10)
u,v∈Λ, u=v
Denote by L (Λ) a class of pseudoperiodic functions s such that a(u) eiut , |a(u)|2 < ∞; s(t) = u∈Λ
(11)
u∈Λ
this class was introduced by Stepanoff (for details, see [5]). It was established in [5] that L (Λ) is a closed subspace of the Banach space L .
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