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Creators/Authors contains: "Pan, Zexin"

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  1. ; (, SIAM Journal on Numerical Analysis)
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  2. ; (, Journal of Complexity)
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  3. ; (, Mathematics of Computation)
    Let f f be analytic on [ 0 , 1 ] [0,1] with | f ( k ) ( 1 / 2 ) | ⩽<#comment/> A α<#comment/> k k ! |f^{(k)}(1/2)|\leqslant A\alpha ^kk! for some constants A A and α<#comment/> > 2 \alpha >2 and all k ⩾<#comment/> 1 k\geqslant 1 . We show that the median estimate of μ<#comment/> = ∫<#comment/> 0 1 f ( x ) d x \mu =\int _0^1f(x)\,\mathrm {d} x under random linear scrambling with n = 2 m n=2^m points converges at the rate O ( n −<#comment/> c log ⁡<#comment/> ( n ) ) O(n^{-c\log (n)}) for any c > 3 log ⁡<#comment/> ( 2 ) / π<#comment/> 2 ≈<#comment/> 0.21 c> 3\log (2)/\pi ^2\approx 0.21 . We also get a super-polynomial convergence rate for the sample median of 2 k −<#comment/> 1 2k-1 random linearly scrambled estimates, when k / m k/m is bounded away from zero. When f f has a p p ’th derivative that satisfies a λ<#comment/> \lambda -Hölder condition then the median of means has error O ( n −<#comment/> ( p + λ<#comment/> ) + ϵ<#comment/> ) O( n^{-(p+\lambda )+\epsilon }) for any ϵ<#comment/> > 0 \epsilon >0 , if k →<#comment/> ∞<#comment/> k\to \infty as m →<#comment/> ∞<#comment/> m\to \infty . The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 
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