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Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(‖x‖),x∈Rn, is the self-convolution of an indicator function supported on the unit ball in Rn. This function is evidently radial and positive definite, and so are its scale mixtures that form the classHn. Our main results characterize the classesHn,n⩾1, andH∞=∩n⩾1Hn. This leads to an analogue of Pólya's criterion for radial functions on Rn,n⩾2: Ifϕ:[0,∞)→R is such thatϕ(0)=1,ϕ(t) is continuous, limt→∞ϕ(t)=0, and(−1)kdkdtk[−ϕ′(t)]is convex fork=[(n−2)/2], the greatest integer less than or equal to (n−2)/2, thenϕ(‖x‖) is a characteristic function in Rn. Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations ofHnto classes of radial positive definite functions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin–Madison), Mittal (Pacific J. Math.64(1976), 517–538), and Berman (Pacific J. Math.78(1978), 1–9), and close with hints at applications in geostatistics.

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We find that the $j$th derivative of the estimators at $x_0$ converges at the rate $n^{-(k-j)/(2k+1)}$ for $j=0,...,k-1$. The limiting distribution depends on an almost surely uniquely defined stochastic process $H_k$ that stays above (below) the $k$-fold integral of Brownian motion plus a deterministic drift when $k$ is even (odd). Both the MLE and LSE are known to be splines of degree $k-1$ with simple knots. Establishing the order of the random gap $\tau_n^+-\tau_n^-$, where $\tau_n^{\pm}$ denote two successive knots, is a key ingredient of the proof of the main results. We show that this ``gap problem'' can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds., Comment: Published in at http://dx.doi.org/10.1214/009053607000000262 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)