By Ghora J.K.

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**Extra info for A Central Limit Theorem for the L2 Error of Positive Wavelet Density Estimator**

**Sample text**

If G is a graph on Q then either G or its complement G complete graph whose vertex set is dense in an interval. Years later this was considerably extended by Galvin and Laver. After a good many somewhat ad hoc results, in 1956 Erd˝ os and Rado gave the ﬁrst systematic treatment of “arrow relations” for cardinals; in their fundamental paper, “A partition calculus in set theory”, they set out to establish a ‘calculus’ of partitions. Among many other results, they proved that if λ ≥ 2 and ρ ≥ ℵ0 are cardinals then (λρ )+ → ((λρ )+ , (ρ+ )ρ )2 , but λρ ((λ · ρ)+ , ρ+ )2 .

It all started with his fascination with Sperner’s theorem on the maximal number of subsets of a ﬁnite set with no subset contained in another. Sperner proved in 1928 that if the ground set has n elements then the maximum is attained by the system of all n/2 subsets. Erd˝ os was quick to appreciate the beauty and importance of this result, and throughout his career frequently returned to problems in this vein. In 1939, Littlewood and Oﬀord gave estimates of the number of real roots of a random polynomial of degree n for various probability spaces of polynomials.

Soon after the turn of the century, Landau, Hardy and Littlewood founded a ﬂourishing branch of analysis by making extensive use of deep results resembling this theorem of Tauber. These Tauberian theorems claim that if a series is summable with a certain method of summation and satisﬁes certain additional conditions then it is also summable with a weaker method of summation. e. limx→∞ e−x sn xn /n! exists, where √ sn = a1 + . . + an ) and nan → 0 then an is convergent. The second part of the elementary proof of the PNT was, essentially, such a Tauberian theorem.