Abstract
We introduce the notion “stochastic quasi-interpolation on compact Hausdorff spaces”, and establish Gaussian-type \(L^p\)-concentration inequalities (\(1 \le p \le \infty \)) for stochastic Bernstein polynomials in terms of the modulus of continuity of a target function \(f \in C[0,1]\). For p in the range \(1 \le p < \infty ,\) these inequalities hold true unconditionally in the sense that no additional assumption on a given target function is required. For the case \(p=\infty \), our proof calls for a crucial application of Dvoretzky–Kiefer–Wolfowitz inequality (Dvoretzky et al. in Ann Math Stat 27(3):642–669, 1956; Massart in Ann Probab 18(3):1269–1283, 1990) , and requires a moderate decay condition on the modulus of continuity. Our result for the case \(p=\infty \) confirms a similar conjecture raised in Sun and Wu (Proc Am Math Soc 147(2):671–679, 2019). As a corollary, we show that for all \(1 \le p \le \infty \) the expected \(L^p\)-approximation order of stochastic Bernstein polynomials is comparable to that given by the classical Bernstein polynomials.
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Notes
Let us take a brief moment off the main topic here to reflect on making a list of standard criteria in evaluating the robustness of a stochastic approximation operator. We suggest it be on the list that the expected approximation power of the stochastic operator matches that of a standard deterministic operator.
The type of exponential decay rates we derive here is often referred to as Gaussian tail bounds.
References
Adell, J.A., Cardenas-Morales, D.: Stochastic Bernstein polynomials: uniform convergence in probability with rates. Adv. Comput. Math. 46, 16 (2020)
Birnbaum, Z.W., McCarty, R.C.: A distribution-free upper confidence bound for \(\text{ Pr }\{Y<X\}\), based on independent samples of \(X\) and \(Y\). Ann. Math. Stat. 29, 558–562 (1958)
Bobkov, S., Ledoux, M.: One-dimensional empirical measures, order statistics and Kantorovich transport distances. Mem. Am. Math. Soc. 261, 1259 (2019)
Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford (2013)
Bourgain, J., Lindenstrauss, J.: Distribution of points on spheres and approximation by zonotopes. Israel J. Math. 64, 25–31 (1988)
Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Math. 162, 73–141 (1989)
Bourgain, J., Lindenstrauss, J.: Approximating the ball by a Minkowski sum of segments with equal length. Discret. Comput. Geom. 9, 131–144 (1993)
Buldygin, V.V., Kozachenko, I.V.: Metric Characterization of Random Variables and Random Processes, Translations of Mathematical Monographs, vol. 188. American Mathematical Society, Providence (2000)
Casella, G., Berger, R.: Statistical Reference, 2nd ed, Duxbury Advanced Series (2002)
Cenusa, Gh., Sacuiu, I.: On some stochastic approximations for random functions. Rend. Mat. (Roma) Ser. VI 12, 143–156 (1979)
Cheney, E.W.: Introduction to Approximation Theory, 2nd edn. Chelsea, New York (1982)
Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. (New Ser.) Am. Math. Soc. 39, 1–49 (2001)
Cucker, F., Zhou, D.X.: Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge (2007)
David, H.A., Nagaraja, H.N.: Order Statistics, Wiley Series in Probability and Statistics, 3rd edn. Wiley, Hoboken (2003)
Ditzian, Z., Ivanov, K.G.: Strong converse inequalities. J. Anal. Math. 61, 61–111 (1993)
Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, Berlin (2012)
Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27(3), 642–669 (1956)
Gal, S.G.: Jackson type estimates in the approximation of random functions by random polynomials. Rend. Mat. Appl. Roma 4, 543–556 (1994)
Gal, S.G.: Approximation Theory in Random Setting, Chapter 12 in Handbook of Analytic Computational Methods in Applied Mathematics, pp. 571–616. Chapman and Hall, Boca Raton (2000)
Gal, S.G., Niculescu, P.C.: Approximation of random functions by stochastic Bernstein polynomials in capacity spaces. Carpathian J. Math. 37(2), 185–194 (2021)
Gao, W., Sun, X., Wu, Z., Zhou, X.: Multivariate Monte Carlo approximation based on scattered data. SIAM J. Sci. Comput. 42, 2262–2280 (2020)
Gavrea, I., Ivan, M.: The Bernstein Voronovskaja-type theorem for positive linear operators. J. Approx. Theory 192, 291–296 (2015)
Ignatov, Z.G., Mills, T.M., Tzankova, I.P.: On the rate of approximation of random functions. Serdica Bulg. Math. Publ. 18, 240–247 (1992)
Kamolov, A.I.: On exact estimates of approximation of random processes (in Russian). Dokl. Akad. Nauk UzSSR 11, 4–6 (1986)
Lorentz, G.G.: Bernstein Polynomials, Mathematical Expositions, vol. 8. University of Toronto Press, Toronto (1953)
Massart, P.: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18(3), 1269–1283 (1990)
Niederrreiter, H.: On the existence of uniformly distributed sequences in compact spaces. Compos. Math. 25(1), 93–99 (1972)
Marchal, O., Arbel, J.: On the sub-Gaussianity of the Beta and Dirichlet distributions. Electron. Commun. Probab. 22(54), 1–14 (2017)
Onicescu, O., Istratescu, V.I.: Approximation theorems for random functions. Rend. Mat. (Roma) Ser. VI 8, 65–81 (1975)
Onicescu, O., Istratescu, V.I.: Approximation theorems for random functions. II. Rend. Mat. (Roma) Ser. VI 11(4), 585–589 (1978)
Sikkema, P.C.: Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, (German). Numer. Math. 3, 107–116 (1961)
Sikkema, P.C.: Über den Grad der approximation mit Bernstein-Polynomen (German). Numer. Math. 1, 221–239 (1959)
Sun, X., Wu, Z.: Chebyshev type inequality for stochastic Bernstein polynomials. Proc. Am. Math. Soc. 147(2), 671–679 (2019)
Sun, X., Wu, Z., Zhou, X.: On probabilistic convergence rates of stochastic Bernstein polynomials. Math. Comp. 90(328), 813–830 (2021)
Totik, V.: Approximation by Bernstein polynomials. Am. J. Math. 116, 995–1018 (1994)
Vitale, R.A.: A Bernstein polynomial approach to density function estimation, statistical inference and related topics 2. In: Proceedings of the Summer Research Institute on Statistical Inference for Stochastic Processes, Bloomington, Indiana, July 31–August 9, pp. 87–99 (1975)
Wagner, G.: On a new method for constructing good point sets on spheres. Discret. Comput. Geom. 9, 111–129 (1993)
Wu, Z., Sun, X., Ma, L.: Sampling scattered data with Bernstein polynomials: stochastic and deterministic error estimates. Adv. Comput. Math. 38, 187–205 (2013)
Wu, Z., Zhou, X.: Polynomial convergence order of stochastic Bernstein approximation. Adv. Comput. Math. 46, 25 (2020)
Acknowledgements
We thank Professor Zongmin Wu, Fudan University, for fruitful discussions on topics related to the article.
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Sun, X., Zhou, X. Stochastic Quasi-Interpolation with Bernstein Polynomials. Mediterr. J. Math. 19, 240 (2022). https://doi.org/10.1007/s00009-022-02150-y
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DOI: https://doi.org/10.1007/s00009-022-02150-y
Keywords
- Dvoretzky–Kiefer–Wolfowitz inequality
- modulus of continuity
- order statistics
- stochastic quasi-interpolation