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Bernstein polynomials and Brownian motion. (English) Zbl 1149.60051

The author studies various aspects of possible connections between random Bernstein polynomials and Brownian motion. Consider, e.g., the random polynomials \[ \mathfrak B_n(x) =\sum_{j=0}^n \binom nj B\left(\frac{j}{n}\right)x^j (1-x)^{n-j},\quad x\in [0,1],\quad n=1,2,\dots, \] where \(\{B(t),0\leq t\leq 1\}\) is a Brownian motion on \([0,1].\) Then, knowing that Brownian motion exists, some properties are studied by means of \(\mathfrak B_n\) such as nondifferentiability of Brownian paths or zeros of random Bernstein polynomials. On the other hand, forgetting about existence of Brownian motion, the question is investigated whether or not polynomials of the type \[ \mathcal B_n(x) =\sum_{j=0}^n \binom nj X_{nj} x^j (1-x)^{n-j},\quad x\in [0,1],\quad n=1,2,\dots, \] with suitable random variables \(X_{nj},\) converge (in a suitable sense) to a Brownian motion. Rigorous proofs are given which should be understandable to a reader with some background in probability.

MSC:

60J65 Brownian motion
12E10 Special polynomials in general fields
60F17 Functional limit theorems; invariance principles
60G15 Gaussian processes
60G17 Sample path properties
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