and the convergence is uniform in .
as .
This section is adapted from: Heads or Tails, by Emmanuel Lesigne, Student Mathematical Library Volume 28, American Mathematical Society, Providence, 2005, Chapter 5
Proposition 1 (Markov’s Inequality) Let be a random variable taking only non-negative values. Then for each
Proof 1
Proposition 2 (Chebyshev’s Inequality) Let be a random variable. Then for
Proof 2
This immediately follows from Markov’s inequality applied to the non-negative random variable . Q.E.D.
and the convergence is uniform in .
Proof 3
The variance of the random variable is . Rewrite the probability as the equivalent event:
By Chebyshev’s inequality
Since , the proof is complete. Q.E.D.
Note that the proof demonstrates that
uniformly in .
Theorem 2 Let be a real function that is defined and continuous on the interval . Then
as .
Proof 4
Not the second application of the triangle inequality on the second summation.
This shows that can be made arbitrarily small, uniformly with respect to , by picking sufficiently large.
Remark 1 The family of polynomials
are called the Bernstein polynomials. The Bernstein polynomials have several useful properties:
Corollary 1 The polynomial uniformly approximating the continuous function on the interval is
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