From Osher Doctorow
Kai Lai Chung (Stanford U.) in his A Course in Probability Theory,
Harcourt, Brace & World: N.Y. 1968, has an exercise on p. 140 that
introduces the "Levy concentration function" Q_F of cdf FX or F,
defined by:
1) Q_V(h) = sup[F(x+h) - F(x-)]
where x- is the limit from the left. It turns out that this sup
(supremum or l.u.b.) is attained and that for G a second cdf, we have:
2) Q_(F*G)(h) < = Q_F(h) ^ Q_G(h)
where ^ is the "minimum of" and * is the convolution.
Chung also has another problem on Levy concentration functions on page
164 which asks one to prove that if Q_n is the concentration function
of Sn = sum X_j (j = 1 to n) for independent identically distributed Xj
random variables with common nondegenerate cdf F or FX, then for every
h > 0 we have:
3) Q_n(h) < = An^(-1/2)
Osher Doctorow
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