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.Comment: 16 page

Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations

In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1002.4546 by other author

Toeplitz Lemma, Complete Convergence and Complete Moment Convergence

In this paper, we study the Toeplitz lemma, the Ces\`{a}ro mean convergence theorem and the Kronecker lemma. At first, we study "complete convergence" versions of the Toeplitz lemma, the Ces\`{a}ro 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 Ces\`{a}ro 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 Ces\`{a}ro mean convergence theorem, and the Kronecker lemma under these two classes of complete moment convergence.Comment: 16 page

Two Stronger Versions of the Union-closed Sets Conjecture

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. In this paper, we introduce two stronger versions of Frankl's conjecture and give a partial proof. Three related questions are introduced.Comment: 26 pages; a typo on Page 23 was revise

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 Vall\'{e}e 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}.Comment: 12 page

Jensen's Inequality for Backward SDEs Driven by GG-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.Comment: 11 page

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.Comment: 15 page

A Note on Uniform Integrability of Random Variables in a Probability Space and Sublinear Expectation Space

In this note we discuss uniform integrability of random variables. In a probability space, we introduce two new notions on uniform integrability of random variables, and prove that they are equivalent to the classic one. In a sublinear expectation space, we give de La Vall\'ee Poussin criterion for the uniform integrability of random variables and do some other discussions.Comment: 10 page

Convergences of Random Variables under Sublinear Expectations

In this note, we will survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, we will prove that $L^p$ convergence is stronger than convergence in capacity, convergence in capacity is stronger than convergence in distribution, and give some equivalent characterizations of convergence in distribution. In addition, we give a dominated convergence theorem under sublinear expectations, which may have its own interest.Comment: 17 page

The 6-element case of S-Frankl conjecture (I)

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. In [3], a stronger version of Frankl's conjecture (S-Frankl conjecture for short) was introduced and a partial proof was given. In particular, it was proved in \cite{CH17} that S-Frankl conjecture holds when $n\leq 5$, where $n$ is the number of all the elements in the family of sets. Now, we want to prove that it holds when $n=6$. Since the paper is very long, we split it into two parts. This is the first part.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1711.0427