139 research outputs found
Hunt's Hypothesis (H) for the Sum of Two Independent Levy Processes
Which Levy processes satisfy Hunt's hypothesis (H) is a long-standing open
problem in probabilistic potential theory. The study of this problem for
one-dimensional Levy processes suggests us to consider (H) from the point of
view of the sum of Levy processes. In this paper, we present theorems and
examples on the validity of (H) for the sum of two independent Levy processes.
We also give a novel condition on the Levy measure which implies (H) for a
large class of one-dimensional Levy processes
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 and 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
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
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
Jensen's Inequality for Backward SDEs Driven by -Brownian motion
In this note, we consider Jensen's inequality for the nonlinear expectation
associated with backward SDEs driven by -Brownian motion (-BSDEs for
short). At first, we give a necessary and sufficient condition for -BSDEs
under which one-dimensional Jensen inequality holds. Second, we prove that for
, the -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 -BSDEs under which the
-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 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
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