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We consider the problem of evaluating I(\varphi):=\int_{\ui^s}\varphi(x) dx for a function ΟβL2[0,1)s. In situations where I(Ο) can be approximated by an estimate of the form Nβ1βn=0Nβ1βΟ(xn), with {xn}n=0Nβ1β a point set in [0,1)s, it is now well known that the OPβ(Nβ1/2) Monte Carlo convergence rate can be improved by taking for {xn}n=0Nβ1β the first N=Ξ»bm points, Ξ»β{1,β¦,bβ1}, of a scrambled (t,s)-sequence in base bβ₯2. In this paper we derive a bound for the variance of scrambled net quadrature rules which is of order o(Nβ1) without any restriction on N. As a corollary, this bound allows us to provide simple conditions to get, for any pattern of N, an integration error of size oPβ(Nβ1/2) for functions that depend on the quadrature size N. Notably, we establish that sequential quasi-Monte Carlo (M. Gerber and N. Chopin, 2015, \emph{J. R. Statist. Soc. B}, 77 (3), 509-579) reaches the oPβ(Nβ1/2) convergence rate for any values of N. In a numerical study, we show that for scrambled net quadrature rules we can relax the constraint on N without any loss of efficiency when the integrand Ο is a discontinuous function while, for sequential quasi-Monte Carlo, taking N=Ξ»bm may only provide moderate gains
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