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Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences
Probability · Submitted 31 May 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities.
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One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process $X(t)$. In his seminal paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6), 801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the prediction error for discrete-time deterministic processes in the following two cases: (a) the spectral density $f(\lambda)$ of $X(t)$ is continuous and vanishes on an interval, (b) the spectral density $f(\lambda)$ has a very high order contact with zero. He showed that in the case (a) the prediction error variance behaves exponentially, while in the case (b), it behaves hyperbolically as $n\to\infty$. In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. | background 1.00 | 0.04 | 0.03 | asserted 0.98 |
| s1 | This behavior depends on the regularity (deterministic or non-deterministic) of the process $X(t)$. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s2 | In his seminal paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6), 801-810, 1957), M. | background 1.00 | 0.03 | 0.02 | asserted 0.53 |
| s3 | Rosenblatt has described the asymptotic behavior of the prediction error for discrete-time deterministic processes in the following two cases: (a) the spectral density $f(\lambda)$ of $X(t)$ is continuous and vanishes on an interval, (b) the spectral density $f(\lambda)$ has a very high order contact with zero. | background 1.00 | 0.02 | 0.05 | asserted 1.00 |
| s4 | He showed that in the case (a) the prediction error variance behaves exponentially, while in the case (b), it behaves hyperbolically as $n\to\infty$. | background 0.90 | 0.04 | 0.06 | asserted 1.00 |
| s5 | In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities. | contribution 1.00 | 0.94 | 0.60 | asserted 1.00 |
| s6 | Examples illustrate the obtained results. | contribution 0.62 | 0.79 | 0.06 | asserted 0.96 |
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Original abstract
One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process $X(t)$. In his seminal paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6), 801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the prediction error for discrete-time deterministic processes in the following two cases: (a) the spectral density $f(\lambda)$ of $X(t)$ is continuous and vanishes on an interval, (b) the spectral density $f(\lambda)$ has a very high order contact with zero. He showed that in the case (a) the prediction error variance behaves exponentially, while in the case (b), it behaves hyperbolically as $n\to\infty$. In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.
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