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.     

On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t={\sum }_{s\le t}F(X_{s?},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t,{\mathbb{P}}_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\infty }$ implies finiteness of its expectation. This result is then applied to study the relative entropy of ${\mathbb{P}}_x$ and the probability measure induced by a purely discontinuous Girsanov transform of the process $X$. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.     

On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes

We study subexponential tail asymptotics for the distribution of the maximum $M_t?{sup}_{u\in [0,t]}{X}_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both processes we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes with drift particularly includes the Cramér–Lundberg renewal risk process.     

Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

The paper considers a process $Z_t=(X_t,Y_t)$ where $X_t$ is the position of a particle and $Y_t$ its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically $\beta$-mixing. In this context, we propose an adaptive non-parametric kernel estimator of the stationary density $p$ of $Z$, based on $n$ discrete time observations with time step $\delta$. Two observation schemes are considered: in the first one, $Z$ is the observed process, in the second one, only $X$ is measured. Estimators are proposed in both settings and upper risk bounds of the mean integrated squared error (MISE) are proved and discussed in each case, the second one being more difficult than the first one. We propose a data driven bandwidth selection procedure based on the Goldenshluger and Lespki (2011) method. In both cases of complete and partial observations, we can prove a bound on the MISE asserting the adaptivity of the estimator. In practice, we take advantage of a very recent improvement of the Goldenshluger and Lespki (2011) method provided by Lacour et al. (2016), which is computationally efficient and easy to calibrate. We obtain convincing simulation results in both observation contexts.     

Domination and upper domination of direct product graphs

Let $X_{\mathbb{Z}∕n\mathbb{Z}}$ denote the unitary Cayley graph of $\mathbb{Z}∕n\mathbb{Z}$. We present results on the tightness of the known inequality $\gamma \left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le {\gamma }_t\left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le g\left(n\right)$, where $\gamma$ and${\gamma }_t$ denote the domination number and total domination number, respectively, and $g$ is the arithmetic function known as Jacobsthal’s function. In particular, we construct integers $n$ with arbitrarily many distinct prime factors such that $\gamma \left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le {\gamma }_t\left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le g\left(n\right)?1$. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.     

Ergodic properties of generalized Ornstein–Uhlenbeck processes

We investigate ergodic properties of the solution of the SDE $d{V}_t=V_{t?}d{U}_t+d{L}_t$, where $(U,L)$ is a bivariate Lévy process. This class of processes includes the generalized Ornstein–Uhlenbeck processes. We provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster–Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.     

Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times ${\tau }_u=inf\{n:Y_n>u\}$ for the perpetuity sequence

$Y_n=B_1+A_1{B}_2+?+(A_1\dots A_{n?1})B_n,$

where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb{R}}^{+}\times \mathbb{R}$. Recently, a number of limit theorems related to ${\tau }_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems (see Buraczewski et al., in press). We obtain a precise asymptotics of the sequence $\mathbb{P}[{\tau }_u=logu∕\rho ]$, $\rho >0$, $u\to \infty$ which considerably improves the previous results of Buraczewski et al. (in press). There, probabilities $\mathbb{P}[{\tau }_u\in I_u]$ were identified, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $loglogu$. Remarkable analogies and differences to random walks (Buraczewski and Ma?lanka, in press; Lalley, 1984) are discussed.     

Lower bounds for moments of global scores of pairwise Markov chains

Let $X_1,\dots$ and $Y_1,\dots$ be random sequences taking values in a finite set $\mathbb{A}$. We consider a similarity score $L_n?L(X_1,\dots ,X_n;Y_1,\dots ,Y_n)$ that measures the homology of words $(X_1,\dots ,X_n)$ and $(Y_1,\dots ,Y_n)$. A typical example is the length of the longest common subsequence. We study the order of moment $E{|L_n?E{L}_n|}^r$ in the case where the two-dimensional process $(X_1,Y_1),(X_2,Y_2),\dots$ is a Markov chain on $\mathbb{A}\times \mathbb{A}$. This general model involves independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a condition that guarantees that $E{|L_n?E{L}_n|}^r?n^{\frac{r}{2}}$. We also perform simulations indicating the validity of the condition.     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution $(Y^{t,\xi },Z^{t,\xi },H^{t,\xi })$, $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ of the split equations are proved. The first and the second order derivatives of the process $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ with respect to $x$, the derivative of the process $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ with respect to the measure $P_{\xi }$, and the derivative of the process $({?}_{\mu }{Y}^{t,x,P_{\xi }}\left(y\right),{?}_{\mu }{Z}^{t,x,P_{\xi }}\left(y\right),{?}_{\mu }{H}^{t,x,P_{\xi }}\left(y\right))$ with respect to $y$ are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in $L^2$. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function $V(t,x,P_{\xi })?Y_t^{t,x,P_{\xi }}$ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.     

Stochastic Komatu–Loewner evolutions and BMD domain constant

For Komatu–Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of Schramm (2000) to find the SDEs satisfied by the induced motion $\xi \left(t\right)$ on $?\mathbb{H}$ and the slit motion $\mathbf{s}\left(t\right)$. The diffusion coefficient $\alpha$ and drift coefficient $b$ of such SDEs are homogeneous functions.Next with solutions of such SDEs, we study the corresponding stochastic Komatu–Loewner evolution, denoted as ${\mathrm{SKLE}}_{\alpha ,b}$. We introduce a function $b_{\mathrm{BMD}}$ measuring the discrepancy of a standard slit domain from $\mathbb{H}$ relative to BMD. We show that ${\mathrm{SKLE}}_{\sqrt{6},?b_{\mathrm{BMD}}}$ enjoys a locality property.     

Sampling in a weighted Sobolev space

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${\{f(t_n)\}}_{n\in \mathbb{Z}}\cup {\{\stackrel{?}{f}({\lambda }_k)\}}_{k\in \mathbb{Z}}$ along appropriate slowly increasing sequences ${\{t_n\}}_{n\in \mathbb{Z}}$ and ${\{{\lambda }_n\}}_{n\in \mathbb{Z}}$ tending to ±∞ as $n\to \pm \infty$.     

Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.     

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.