By Blau W.

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1 ≥ P(Ωn ) = ω∈Ωn Therefore, |Ωn | ≤ en(H+ε) . On the other hand, for suﬃciently large n ε 1 ≤ P(Ωn ) ≤ e−n(H− 2 ) |Ωn |, 2 ε and therefore |Ωn | ≥ 12 en(H− 2 ) ≥ en(H−ε) for suﬃciently large n. 7. Let f be a continuous function on the closed interval [0, 1]. For every ε > 0 there exists a polynomial bn (x) of degree n such that max |bn (x) − f (x)| ≤ ε. 0≤x≤1 The proof of this theorem, which we present now, is due to S. Bernstein. Proof. Consider the function n bn (x) = k=0 n! (n − k)! n which is called the Bernstein polynomial of the function f .

This completes the proof of the theorem. ✷ The next theorem is referred to as the Law of Large Numbers for a Homogeneous Sequence of Independent Trials. 5. For any ε > 0, P(| νn − p| < ε) → 1 as n → ∞. n Proof. By the Chebyshev Inequality, P(| ≤ νn − p| ≥ ε) = P(|ν n − np| ≥ nε) n np(1 − p) p(1 − p) Var(ν n ) = = → 0 as n → ∞. 2 2 2 2 n ε n ε nε2 ✷ 28 2 Sequences of Independent Trials The Law of Large Numbers states that for a homogeneous sequence of independent trials, typical realizations are such that the frequency with which an event B appears in ω is close to the probability of this event.

Conversely, any probability measure µ on the Borel sets of the real line deﬁnes a distribution function via the formula F (x) = µ((−∞, x]). Thus there is a one-to-one correspondence between probability measures on the real line and distribution functions. 10. Similarly, there is a one-to-one correspondence between the distribution functions on Rn and the probability measures on the Borel sets of Rn . , xn ) = µ((−∞, x1 ] × ... × (−∞, xn ]). Example. Let f be a function deﬁned on an interval [a, b] of the real line.