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result(s) for
"Random variables"
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Dynamics of the Box-Ball System with Random Initial Conditions via Pitman’s Transformation
The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In
this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al.
recently showed the invariance in distribution of Bernoulli product measures with density strictly less than
eBook
The Drunkard's walk : how randomness rules our lives
An irreverent look at how randomness influences our lives, and how our successes and failures are far more dependent on chance events than we recognize.
Book
On normal approximation for φ-mixing and m-dependent random variables
In this paper, we estimate the difference |
E
h
(
Z
n
)
−
E
h
(
Y
)
|
between the expectations of real finite Lipschitz function
h
of the sum
Z
n
= (
X
1
+ ⋯ +
X
n
)
/B
n
, where
B
n
2
=
E
(
X
1
+ ⋯ +
X
n
)
2
>
0, and a standard normal random variable
Y
, where real centered random variables
X
1
,X
2
,
… satisfy the
φ
-mixing condition, defined between the “past” and “ future”, or are
m
-dependent. In particular cases, under the condition
∑
r
=
1
∞
r
φ
(
r
)
<
∞
or
∑
r
=
1
∞
r
φ
1
/
2
(
r
)
<
∞
, the obtained upper bounds for
φ
-mixing random variables are of order
O
(
n
−
1
/
2
). In addition, we refine the previously known upper bounds of order
O
((
m
+ 1)
1+
δ
L
2+
δ,n
), where
L
2+
δ,n
is the Lyapunov fraction of order 2 +
δ
, for
m
-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.
Journal Article
Fooled by randomness : the hidden role of chance in life and in the markets
This is a book about luck. More specifically, it is a book about how we perceive luck, twist it around and regard it as intention or purpose.
Book
On the rate of convergence in Wasserstein distance of the empirical measure
Let
μ
N
be the empirical measure associated to a
N
-sample of a given probability distribution
μ
on
R
d
. We are interested in the rate of convergence of
μ
N
to
μ
, when measured in the Wasserstein distance of order
p
>
0
. We provide some satisfying non-asymptotic
L
p
-bounds and concentration inequalities, for any values of
p
>
0
and
d
≥
1
. We extend also the non asymptotic
L
p
-bounds to stationary
ρ
-mixing sequences, Markov chains, and to some interacting particle systems.
Journal Article
Higher moments of Banach space valued random variables
We define the
We study both the projective and injective
tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations:
Bochner integrals, Pettis integrals and Dunford integrals.
One of the problems studied is whether two random variables with the
same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show
that this holds if the Banach space has the approximation property, but not in general.
Several chapters are devoted to results
in special Banach spaces, including Hilbert spaces,
One of the main motivations of this paper is the application to Zolotarev metrics and
their use in the contraction method. This is sketched in an appendix.
eBook
Nonparametric Bayes Modeling of Populations of Networks
Replicated network data are increasingly available in many research fields. For example, in connectomic applications, interconnections among brain regions are collected for each patient under study, motivating statistical models which can flexibly characterize the probabilistic generative mechanism underlying these network-valued data. Available models for a single network are not designed specifically for inference on the entire probability mass function of a network-valued random variable and therefore lack flexibility in characterizing the distribution of relevant topological structures. We propose a flexible Bayesian nonparametric approach for modeling the population distribution of network-valued data. The joint distribution of the edges is defined via a mixture model that reduces dimensionality and efficiently incorporates network information within each mixture component by leveraging latent space representations. The formulation leads to an efficient Gibbs sampler and provides simple and coherent strategies for inference and goodness-of-fit assessments. We provide theoretical results on the flexibility of our model and illustrate improved performance-compared to state-of-the-art models-in simulations and application to human brain networks. Supplementary materials for this article are available online.
Journal Article
An active learning kriging model for hybrid reliability analysis with both random and interval variables
Hybrid reliability analysis (HRA) with both random and interval variables is investigated in this paper. Firstly, it is figured out that a surrogate model just rightly predicting the sign of performance function can meet the requirement of HRA in accuracy. According to this idea, a methodology based on active learning Kriging (ALK) model named ALK-HRA is proposed. When constructing the Kriging model, the presented method only finely approximates the performance function in the region of interest: the region where the sign tends to be wrongly predicted. Based on the constructed Kriging model, Monte Carlo Simulation (MCS) is carried out to estimate both the lower and upper bounds of failure probability. ALK-HRA is accurate enough with calling the performance function as few times as possible. Four numerical examples and one engineering application are investigated to demonstrate the performance of the proposed method.
Journal Article
Hermite–Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications
In the article, we present several Hermite–Hadamard type inequalities for the co-ordinated convex and quasi-convex functions and give an application to the product of the moment of two continuous and independent random variables. Our results are generalizations of some earlier results. Additionally, an illustrative example on the probability distribution is given to support our results.
Journal Article