Let f(x) be a continuous function defined for 0 ≤ x ≤ 1. Consider the functions
(called Bernstein polynomials) and prove that
Let X1, X2, . . . be independent Bernoulli random variables with mean x. Show that
Bn(x) = E[f(X1 + · · · + Xn / n)]
and then use Theoretical Exercise 4.
Since it can be shown that the convergence of Bn(x) to f (x) is uniform in x, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.

  • CreatedOctober 22, 2015
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