# THE HOFFMANN-JORGENSEN INEQUALITY IN METRIC SEMIGROUPS

@article{Khare2017THEHI, title={THE HOFFMANN-JORGENSEN INEQUALITY IN METRIC SEMIGROUPS}, author={Apoorva Khare and Bala Rajaratnam}, journal={Annals of Probability}, year={2017}, volume={45}, pages={4101-4111} }

We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in the Banach space literature, including those by Johnson and Schechtman Ann. Probab. 17 (1989) 789-808], Klass and Nowicki Ann. Probab. 28 (2000) 851-862], and Hitczenko and Montgomery-Smith Ann. Probab. 29 (2001) 447-466]. Finally, we show that the Hoffmann…

## 6 Citations

Probability inequalities and tail estimates for metric semigroups

- Mathematics
- 2015

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in…

THE KHINCHIN–KAHANE INEQUALITY AND BANACH SPACE EMBEDDINGS FOR ABELIAN METRIC GROUPS

- Mathematics
- 2018

The Khinchin–Kahane inequality is a fundamental result in the probability literature. The most general version to date holds in Banach spaces. Modern applications however necessitate extending this…

The Khinchin-Kahane inequality and Banach space embeddings for metric groups

- Mathematics
- 2016

We extend the Khinchin-Kahane inequality to an arbitrary abelian metric group $\mathscr{G}$. In the special case where $\mathscr{G}$ is normed, we prove a refinement which is sharp and which extends…

Homogeneous length functions on groups

- Mathematics
- 2018

A pseudo-length function defined on an arbitrary group G “ pG, ̈, e, p qq is a map l : G Ñ r0,`8q obeying lpeq “ 0, the symmetry property lpxq “ lpxq, and the triangle inequality lpxyq ď lpxq ` lpyq…

Homogeneous length functions on groups

- MathematicsAlgebra & Number Theory
- 2018

A pseudolength function defined on an arbitrary group G â� (G, Â·, e, ( ) â��1 ) is a map â��: G â�� 0, +â��) obeying â��(e) â� 0, the symmetry property â��(x â��1 ) â� â��(x), and the triangle…

Homogeneous Length Functions on Groups: Intertwined Computer and Human Proofs

- MathematicsJournal of Automated Reasoning
- 2019

A computer generated but human readable proof was read, understood, generalized and abstracted by mathematicians to obtain the key lemma in an interesting mathematical result.

## References

SHOWING 1-10 OF 15 REFERENCES

Probability inequalities and tail estimates for metric semigroups

- Mathematics
- 2015

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in…

The Khinchin-Kahane inequality and Banach space embeddings for metric groups

- Mathematics
- 2016

We extend the Khinchin-Kahane inequality to an arbitrary abelian metric group $\mathscr{G}$. In the special case where $\mathscr{G}$ is normed, we prove a refinement which is sharp and which extends…

An improvement of Hoffmann-Jorgensen's inequality

- Mathematics
- 2000

Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability…

Invariant metrics in groups (solution of a problem of Banach)

- Mathematics
- 1952

Introduction. If G is a semi-group and p a metric on G, p will be called left invariant if p(gx, gy) =p(x, y) whenever {g, x, y} CG, right invariant if always p(xg, yg) =p(x, y), and invariant if it…

Differential calculus on the space of countable labelled graphs

- Mathematics
- 2014

The study of very large graphs is becoming increasingly prominent in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their…

Probability in Banach Spaces: Isoperimetry and Processes

- Mathematics
- 1991

Notation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and…

Integration and measures on the space of countable labelled graphs

- Mathematics
- 2015

In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study…

Large Networks and Graph Limits

- MathematicsColloquium Publications
- 2012

Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks.

Lectures on Differential Geometry

- Mathematics
- 1964

Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations…