A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2({\mathbb{D}}^2)$. Specifically, we study Beurling-type submodules – namely submodules of the form $\theta H^2({\mathbb{D}}^2)$ for θ inner – using properties of Agler decompositions of θ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2({\mathbb{D}}^2)?\theta H^2({\mathbb{D}}^2)$. Results include characterizations (in terms of θ) of when a commutator $[S_{z_j}^{?},S_{z_j}]$ has rank n and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.     

The free energy of the random walk pinning model

We consider the random walk pinning model. This is a random walk on ${\mathbb{Z}}^d$ whose law is given as the Gibbs measure ${\mu }_{N,Y}^{\beta }$, where the polymer measure ${\mu }_{N,Y}^{\beta }$ is defined by using the collision local time with another simple symmetric random walk $Y$ on ${\mathbb{Z}}^d$ up to time $N$. Then, at least two definitions of the phase transitions are known, described in terms of the partition function and the free energy. In this paper, we will show that the two critical points coincide and give an explicit formula for the free energy in terms of a variational representation. Also, we will prove that if $\beta$ is smaller than the critical point, then $X$ under ${\mu }_{N,Y}^{\beta }$ satisfies the central limit theorem and the invariance principle $P_Y$-almost surely.     

Multivariate polynomial interpolation and sampling in Paley–Wiener spaces

In this paper, an equivalence between existence of particular exponential Riesz bases for spaces of multivariate bandlimited functions and existence of certain polynomial interpolants for functions in these spaces is given. Namely, polynomials are constructed which, in the limiting case, interpolate ${\left\{({\tau }_n,f\left({\tau }_n\right))\right\}}_n$ for certain classes of unequally spaced data nodes ${\left\{{\tau }_n\right\}}_n$ and corresponding ${?}_2$ sampled data ${\left\{f\left({\tau }_n\right)\right\}}_n$. Existence of these polynomials allows one to construct a simple sequence of approximants for an arbitrary multivariate bandlimited function $f$ which demonstrates $L_2$ and uniform convergence on ${\mathbb{R}}^d$ to $f$. A simpler computational version of this recovery formula is also given at the cost of replacing $L_2$ and uniform convergence on ${\mathbb{R}}^d$ with $L_2$ and uniform convergence on increasingly large subsets of ${\mathbb{R}}^d$. As a special case, the polynomial interpolants of given ${?}_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinent Riesz bases and unequally spaced data nodes are also given.     

Limit theorems for Hilbert space-valued linear processes under long range dependence

Let ${\left(X_k\right)}_{k\in \mathbb{Z}}$ be a linear process with values in a separable Hilbert space $\mathbb{H}$ given by $X_k={\sum }_{j=0}^{\infty }{(j+1)}^{?N}{\epsilon }_{k?j}$ for each $k\in \mathbb{Z}$, where $N:\mathbb{H}\to \mathbb{H}$ is a bounded, linear normal operator and ${\left({\epsilon }_k\right)}_{k\in \mathbb{Z}}$ is a sequence of independent, identically distributed $\mathbb{H}$-valued random variables with $E{\epsilon }_0=0$ and $E6{\epsilon }_0{6}^2<\infty$. We investigate the central and the functional central limit theorem for ${\left(X_k\right)}_{k\in \mathbb{Z}}$ when the series of operator norms ${\sum }_{j=0}^{\infty }6{(j+1)}^{?N}{6}_{op}$ diverges. Furthermore, we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.     

On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.     

Stochastic comparison of residual lifetime mixture models

In this paper, we consider stochastic comparisons of mixtures ($F^1$ and $G^1$) of residual life distributions ($F_{\theta }$ and $G_{\theta }$, $\theta >0$) arising out of different baseline distributions ($F$ and $G$) and different mixing distributions ($H_1$ and $H_2$). Under certain conditions on $F$, $G$, $H_1$ and $H_2$, we establish that the distributions $F^1$ and $G^1$ are stochastically ordered. Further, we make stochastic comparison of mixture distribution $F^1$ with the distribution of residual lifetime at random time ${\Theta }_1$.     

Scattering for wave maps exterior to a ball

We consider 1-equivariant wave maps from ${\mathbb{R}}_t\times ({\mathbb{R}}_x^3\setminus B)\to S^3$ where $B$ is a ball centered at 0, and $\partial B$ gets mapped to a fixed point on $S^3$. We show that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree.     

Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials

$X_n\left(t\right)=u+\frac{1}{\sqrt{n}}\sum _{k=1}^n(A_{k}cos\left(kt\right)+B_{k}sin\left(kt\right))\text{,}t\in [0,2\pi ]\text{,}u\in \mathbb{R}$

whose coefficients $A_k,B_k$, $k\in \mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a,b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]?[0,2\pi ]$, we prove that

$\underset{n\to \infty }{lim}\frac{\mathbb{E}{N}_n[a,b]}{n}=\frac{b?a}{\pi \sqrt{3}}exp(?\frac{u^2}{2})\text{.}$