Non-Central Limit Theorem for Quadratic Functionals of Hermite-Driven Long Memory Moving Average Processes

Let $(Z_t^{(q, H)})_{t \geq 0}$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. Consider the Hermite-driven moving average process $$X_t^{(q, H)} = \int_0^t x(t-u) dZ^{(q, H)}(u), \qquad t \geq 0.$$ In the special case of $x(u) = e^{-\theta u}, \theta > 0$, $X$ is the non-stationary Hermite Ornstein-Uhlenbeck process of order $q$. Under suitable integrability conditions on the kernel $x$, we prove that as $T \to \infty$, the normalized quadratic functional $$G_T^{(q, H)}(t)=\frac{1}{T^{2H_0 - 1}}\int_0^{Tt}\Big(\big(X_s^{(q, H)}\big)^2 - E\Big[\big(X_s^{(q, H)}\big)^2\Big]\Big) ds , \qquad t \geq 0,$$ where $H_0 = 1 + (H-1)/q$, converges in the sense of finite-dimensional distribution to the Rosenblatt process of parameter $H' = 1 + (2H-2)/q$, up to a multiplicative constant, irrespective of self-similarity parameter whenever $q \geq 2$. In the Gaussian case $(q=1)$, our result complements the study started by Nourdin \textit{et al} in arXiv:1502.03369, where either central or non-central limit theorems may arise depending on the value of self-similarity parameter. A crucial key in our analysis is an extension of the connection between the classical multiple Wiener-It\^{o} integral and the one with respect to a random spectral measure (initiated by Taqqu (1979)), which may be independent of interest.Comment: Accepted for publication in Stoch. Dy

Strongly elliptic pseudodifferential equations on the sphere with radial basis functions

Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and collocation methods. A salient feature of the paper is a {\em unified theory} for error analysis of both approximation methods

On left-orderability and cyclic branched coverings

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for which the r-th cyclic branched covering of S^3 along the knot is left-orderable.Comment: 9 pages, 1 figur

A Common Derivation for Markov Chain Monte Carlo Algorithms with Tractable and Intractable Targets

Markov chain Monte Carlo is a class of algorithms for drawing Markovian samples from high-dimensional target densities to approximate the numerical integration associated with computing statistical expectation, especially in Bayesian statistics. However, many Markov chain Monte Carlo algorithms do not seem to share the same theoretical support and each algorithm is proven in a different way. This incurs many terminologies and ancillary concepts, which makes Markov chain Monte Carlo literature seems to be scattered and intimidating to researchers from many other fields, including new researchers of Bayesian statistics. A generalised version of the Metropolis-Hastings algorithm is constructed with a random number generator and a self-reverse mapping. This formulation admits many other Markov chain Monte Carlo algorithms as special cases. A common derivation for many Markov chain Monte Carlo algorithms is useful in drawing connections and comparisons between these algorithms. As a result, we now can construct many novel combinations of multiple Markov chain Monte Carlo algorithms that amplify the efficiency of each individual algorithm. Specifically, we propose two novel sampling schemes that combine slice sampling with directional or Hamiltonian sampling. Our Hamiltonian slice sampling scheme is also applicable in the pseudo-marginal context where the target density is intractable but can be unbiasedly estimated, e.g. using particle filtering.Comment: Novel designs for multivariate, directional, elliptical and pseudo marginal Hamiltonian slice sampling. This update improved the flow of ideas and clarity up to section 3.6 where major enhancement in the notation, explanation for Neal's recursive proposal generation mechanism. Strong emphasis also on showing that MH-sampling is actually slice sampling in disguis

Reidemeister torsion and Dehn surgery on twist knots

We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a Dehn surgery on a twist knot.Comment: 8 pages, 1 figur

Left-orderability for surgeries on twisted torus knots

We show that the fundamental group of the $3$-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is sufficiently close to $0$.Comment: 6 page

The universal character ring of the (-2,2m+1,2n)-pretzel link

We explicitly calculate the universal character ring of the (-2,2m+1,2n)-pretzel link and show that it is reduced for all integers m and n.Comment: Very minor changes. To appear in the International Journal of Mathematic

On the AJ conjecture for cables of twist knots

We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in $S^3$. We confirm the AJ conjecture for $(r,2)$-cables of the $m$-twist knot, for all odd integers $r$ satisfying $\begin{cases} (r+8)(r-8m)>0 &{if~} m> 0, \\ r(r+8m-4)>0 &{if~} m<0.\end{cases}$Comment: Results in Section 3 are corrected. arXiv admin note: text overlap with arXiv:1111.5258, arXiv:1405.4055; and with arXiv:math/0407521 by other author

Adjoint twisted Alexander polynomials of genus one two-bridge knots

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.Comment: 11 page

Simple Method for Evaluating Singular Integrals

In this paper, we study the class of one dimensional singular integrals that converge in the sense of Cauchy principal value. In addition, we present a simple method for approximating such integrals