For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$...

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $xgt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths,...

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate theof the times, ${\overline{T}}_b\left(r\right)$ and $T_b^{*}\left(r\right)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (two-sided exit), $0\le blt;1$, as $r\downarrow 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are...

A notion of a wide-sense Markov process $X_t$ of order k ≥ 1, $X_t\sim WM\left(k\right)$, is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_t$ is the k-dimensional process $x_t=(X_{t-k+1},...,X_t)$. The covariance structure of $X_t\sim WM\left(k\right)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_t\sim WM\left(k\right)$ iff $x_t$ is a k-dimensional WM(1) process and iff the covariance function of $x_t$ has the triangular...

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider...

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $\left(K\right(t),i(t),Y(t\left)\right)$ on $({𝕋}^2\times \{1,2\}\times {ℝ}^2)$, where ${𝕋}^2$ is the two-dimensional torus. Here $\left(K\right(t),i(t\left)\right)$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y\left(t\right)$ is an additive functional of $K$, defined as ${\int }_0^{t}v\left(K\left(s\right)\right)\mathrm{d}s$, where $\left|v\right|\sim 1$ for small $k$. We prove that the rescaled process ${(NlnN)}^{-1/2}Y\left(Nt\right)$ converges in distribution to a two-dimensional Brownian motion. As a consequence,...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...