Nonparametric quantile estimation using importance sampling

Nonparametric estimation of a quantile of a random variable m(X) is considered, where \(m: \mathbb {R}^d\rightarrow \mathbb {R}\) is a function which is costly to compute and X is a \(\mathbb {R}^d\)-valued random variable with a given density. An importance sampling quantile estimate of m(X), which is based on a suitable estimate \(m_n\) of m, is defined, and it is shown that this estimate achieves a rate of convergence of order \(\log ^{1.5}(n)/n\). The finite sample size behavior of the estimate is illustrated by simulated data.     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.     

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.     

Anderson Polymer in a Fractional Brownian Environment: Asymptotic Behavior of the Partition Function

We consider the Anderson polymer partition function

$$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$

where \(\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}\) is a family of independent fractional Brownian motions all with Hurst parameter \(H\in (0,1)\), and \(\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}\) is a continuous-time simple symmetric random walk on \(\mathbb {Z}^d\) with jump rate \(\kappa \) and started from the origin. \(\mathbb {E}^X\) is the expectation with respect to this random walk. We prove that when \(H\le 1/2\), the function u(t) almost surely grows asymptotically like \(e^{\lambda t}\), where \(\lambda >0\) is a deterministic number. More precisely, we show that as t approaches \(+\infty \), the expression \(\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}\) converges both almost surely and in the \(\hbox {L}^1\) sense to some positive deterministic number \(\lambda \). For \(H>1/2\), we first show that \(\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) exists both almost surely and in the \(\hbox {L}^1\) sense and equals a strictly positive deterministic number (possibly \(+\infty \)); hence, almost surely u(t) grows asymptotically at least like \(e^{\alpha t}\) for some deterministic constant \(\alpha >0\). On the other hand, we also show that almost surely and in the \(\hbox {L}^1\) sense, \(\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)\) is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like \(e^{\beta t\sqrt{\log t}}\) for some deterministic positive constant \(\beta \). Finally, for \(H>1/2\) when \(\mathbb {Z}^d\) is replaced by a circle endowed with a Hölder continuous covariance function, we show that \(\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like \(e^{c t}\) for some deterministic positive constant c.

     

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on \(\mathbb {R}^d\) and entire functions on \(\mathbb {C}\). It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let \(f:\mathbb {R}^d\rightarrow \mathbb {R}^d\) be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates \(f^n\) tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of \(|f^n(x)|\) is asymptotic to the iterated maximum modulus \(M^{n}(R,f)\).     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics

$$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$

which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

Packing measure of super separated iterated function systems

Let J be the limit set of an iterated function system in \(\mathbb {R}^d\) satisfying the open set condition. It is well known that the h-dimensional packing measure of J is positive and finite when h is given by Hutchinson’s formula. However, it may be hard to find a formula for the h-dimensional packing measure of J. We introduce the super separation condition and use it to reduce the problem of computing the packing measure to checking densities of a finite number of balls around each point in the limit set. We then use this fact to find formulas for the packing measure of a class of Cantor sets in \(\mathbb {R}\), a class of fractals based on regular convex polygons in \(\mathbb {R}^2\), and a class of fractals based on regular simplexes in \(\mathbb {R}^d\) for \(d \ge 3\).     

The Minimal Size of Infinite Maximal Antichains in Direct Products of Partial Orders

For a partial order \(\mathbb {P}\) having infinite antichains by \(\mathfrak {a}(\mathbb {P})\) we denote the minimal cardinality of an infinite maximal antichain in \(\mathbb {P}\) and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that \(\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})\), for all \(n\in \mathbb {N}\), where \(\mathfrak {p} (\text {sq} \mathbb {P})\) is the minimal size of a centered family without a lower bound in the separative quotient of the poset \(\mathbb {P}\), or \(\mathfrak {p} (\text {sq} \mathbb {P})=\infty \), if there is no such family. So we have \(\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})\) whenever \(\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})\) and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤ø ₁ by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form \(\langle \mathcal {C} ,\subset \rangle \), where \(\mathcal {C}\) is a family of nonempty closed sets in a compact T ₁-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {A _i ×B _i :i∈I} an infinite partition of the square X ², then at least one of the families {A _i :i∈I} and {B _i :i∈I} contains an infinite partition of X.     

Sharp geometric condition for null-controllability of the heat equation on $$\mathbb {R}^d$$ and consistent estimates on the control cost

In this note we study the control problem for the heat equation on \(\mathbb {R}^d\), \(d\ge 1\), with control set \(\omega \subset \mathbb {R}^d\). We provide a necessary and sufficient condition (called \((\gamma , a)\)-thickness) on \(\omega \) such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the \(\mathbb {R}^d\) case.     

From <Emphasis Type="Italic">r</Emphasis>-dual sets to uniform contractions

Let \(\mathbb {M}^d\) denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of a given set in \(\mathbb {M}^d\) is the intersection of balls of radii r centered at the points of the a given set. In this paper we prove that for any set of given volume in \(\mathbb {M}^d\) the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser–Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. We prove a special case of the Kneser–Poulsen conjecture namely, we prove the conjecture for uniform contractions (with sufficiently large N) in \(\mathbb {M}^d\).     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

Fourier Multiplier Theorems Involving Type and Cotype

In this paper we develop the theory of Fourier multiplier operators \(T_{m}:L^{p}({\mathbb R}^{d};X)\rightarrow L^{q}({\mathbb R}^{d};Y)\), for Banach spaces X and Y, \(1\le p\le q\le \infty \) and \(m:{\mathbb R}^d\rightarrow \mathcal {L}(X,Y)\) an operator-valued symbol. The case \(p=q\) has been studied extensively since the 1980s, but far less is known for \(p<q\). In the scalar setting one can deduce results for \(p<q\) from the case \(p=q\). However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that X and Y are UMD spaces and that m satisfies a smoothness condition. We show that for \(p<q\) other geometric conditions on X and Y, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for \(T_m\) without any smoothness properties of m. Under smoothness conditions the boundedness results can be extrapolated to other values of p and q as long as \(\tfrac{1}{p}-\tfrac{1}{q}\) remains constant.     

Riemann Localisation on the Sphere

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), we mean that for a suitable subset X of \(\mathbb {L}_{p}(\mathbb {S}^{d})\), \(1\le p\le \infty \), the \(\mathbb {L}_{p}\)-norm of the Fourier local convolution of \(f\in X\) converges to zero as the degree goes to infinity. The Fourier local convolution of f at \(\mathbf {x}\in \mathbb {S}^{d}\) is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of \(\mathbf {x}\). The failure of Riemann localisation for \(d>2\) can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

On the Skitovich–Darmois theorem for some locally compact Abelian groups

Let X be a locally compact Abelian group, \(\alpha _{j}, \beta _j\) be topological automorphisms of X. Let \(\xi _1, \xi _2\) be independent random variables with values in X and distributions \(\mu _j\) with non-vanishing characteristic functions. It is known that if X contains no subgroup topologically isomorphic to the circle group \(\mathbb {T}\), then the independence of the linear forms \(L_1=\alpha _1\xi _1+\alpha _2\xi _2\) and \(L_2=\beta _1\xi _1+\beta _2\xi _2\) implies that \(\mu _j\) are Gaussian distributions. We prove that if X contains no subgroup topologically isomorphic to \(\mathbb {T}^2\), then the independence of \(L_1\) and \(L_2\) implies that \(\mu _j\) are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of X generated by an element of order 2. The proof is based on solving the Skitovich–Darmois functional equation on some locally compact Abelian groups.     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.