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Ze-Chun Hu

Publications and source records attributed to Ze-Chun Hu.

48 records · Page 3Linked to original sources
arXiv2016

A Note on Uniform Nonintegrability of Random Variables

In a recent paper \cite{CHR16}, Chandra, Hu and Rosalsky introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. In this note, we introduce a weaker definition on uniform nonintegrability (W-UNI for short) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W-UNI, one of which is a W-UNI analogue of the celebrated de La Vallée Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised in \cite{CHR16}.

math.PR↗
arXiv2015

Further study on Hunt's hypothesis (H) for Levy processes

Getoor's conjecture that essentially all Levy processes satisfy (H) is a long-standing open problem in potential theory. In the beginning of the paper, we summarize the main results obtained so far for the problem. Then, we present two new necessary and sufficient conditions for the validity of (H). Furthermore, we give applications of these new criteria. First, we give explicit constructions of Levy processes satisfying (H) in a context where previously known results could not be applied. Second, we show that a large class of pure jump subordinators can be decomposed into the summation of two independent subordinators such that both of them satisfy (H).

math.PR↗
arXiv2015

Some inequalities and limit theorems under sublinear expectations

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob's inequality for submartingale and Kolmogrov's inequality. By Kolmogrov's inequality, we obtain a special version of Kolmogrov's law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.

math.PR↗
arXiv2015

Toeplitz Lemma, Complete Convergence and Complete Moment Convergence

In this paper, we study the Toeplitz lemma, the Cesàro mean convergence theorem and the Kronecker lemma. At first, we study "complete convergence" versions of the Toeplitz lemma, the Cesàro mean convergence theorem and the Kronecker lemma. Two counterexamples show that they can fail in general and some sufficient conditions for "complete convergence" version of the Cesàro mean convergence theorem are given. Secondly we introduce two classes of complete moment convergence, which are stronger versions of mean convergence and consider the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma under these two classes of complete moment convergence.

math.PR↗
arXiv2014

On variants of conflict-free-coloring for hypergraphs

Conflict-free coloring is a kind of vertex coloring of hypergraphs requiring each hyperedge to have a color which appears only on one vertex. More generally, for a positive integer $k$ there are $k$-conflict-free colorings ($k$-CF-colorings for short) and $k$-strong-conflict-free colorings ($k$-SCF-colorings for short). %for some positive integer $k$. Let $H_n$ be the hypergraph of which the vertex-set is $V_n=\{1,2,\dots,n\}$ and the hyperedge-set $\cal{E}_n$ is the set of all (non-empty) subsets of $V_n$ consisting of consecutive elements of $V_n$. Firstly, we study the $k$-SCF-coloring of $H_n$, give the exact $k$-SCF-coloring chromatic number of $H_n$ for $k=2,3$, and present upper and lower bounds of the $k$-SCF-coloring chromatic number of $H_n$ for all $k$. Secondly, we give the exact $k$-CF-coloring chromatic number of $H_n$ for all $k$.

math.CO↗
arXiv2014

Jensen's Inequality for Backward SDEs Driven by $G$-Brownian motion

In this note, we consider Jensen's inequality for the nonlinear expectation associated with backward SDEs driven by $G$-Brownian motion ($G$-BSDEs for short). At first, we give a necessary and sufficient condition for $G$-BSDEs under which one-dimensional Jensen inequality holds. Second, we prove that for $n>1$, the $n$-dimensional Jensen inequality holds for any nonlinear expectation if and only if the nonlinear expectation is linear, which is essentially due to Jia (Arch. Math. 94 (2010), 489-499). As a consequence, we give a necessary and sufficient condition for $G$-BSDEs under which the $n$-dimensional Jensen inequality holds.

math.PR↗
arXiv2014

New results on Hunt's hypothesis (H) for Lévy processes

In this paper, we present new results on Hunt's hypothesis (H) for Lévy processes. We start with a comparison result on Lévy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the Kanda-Forst-Rao theorem, we give examples of subordinators satisfying (H). Afterwards we give a new necessary and sufficient condition for (H) and obtain an extended Kanda-Forst-Rao theorem. By virtue of this theorem, we give a new class of Lévy processes satisfying (H). Finally, we construct a type of subordinators that does not satisfy Rao's condition.

math.PR↗
arXiv2012

Hunt's hypothesis (H) and Getoor's conjecture for Lévy Processes

In this paper, Hunt's hypothesis (H) and Getoor's conjecture for Lévy processes are revisited. Let $X$ be a Lévy process on $\mathbf{R}^n$ with Lévy-Khintchine exponent $(a,A,μ)$. {First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $μ({\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n})<\infty$, we show that $X$ satisfies (H) if and only if the equation $$ \sqrt{A}y=-a-\int_{\{x\in {\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n}:\,|x|<1\}}xμ(dx),\ y\in \mathbf{R}^n, $$ has at least one solution. Finally, we show that if $X$ is a subordinator and satisfies (H) then its drift coefficient must be 0.}

math.PR↗
arXiv2012

On joint ruin probabilities of a two-dimensional risk model with constant interest rate

In this note we consider the two-dimensional risk model introduced in Avram et al. \cite{APP08} with constant interest rate. We derive the integral-differential equations of the Laplace transforms, and asymptotic expressions for the finite time ruin probabilities with respect to the joint ruin times $T_{\rm max}(u_1,u_2)$ and $T_{\rm min}(u_1,u_2)$ respectively.

math.PR↗