The Annals of Probability

The Hoffmann–Jørgensen inequality in metric semigroups

Apoorva Khare and Bala Rajaratnam

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Abstract

We prove a refinement of the inequality by Hoffmann–Jørgensen 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–Jørgensen inequality (including our generalized version) holds not only in Banach spaces but more generally, in a very primitive mathematical framework required to state the inequality: a metric semigroup $\mathscr{G}$. This includes normed linear spaces as well as all compact, discrete or (connected) abelian Lie groups.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 4101-4111.

Dates
Received: May 2016
Revised: October 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773673

Digital Object Identifier
doi:10.1214/16-AOP1160

Mathematical Reviews number (MathSciNet)
MR3729624

Zentralblatt MATH identifier
06838116

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords
Hoffmann–Jørgensen inequality metric semigroup

Citation

Khare, Apoorva; Rajaratnam, Bala. The Hoffmann–Jørgensen inequality in metric semigroups. Ann. Probab. 45 (2017), no. 6A, 4101--4111. doi:10.1214/16-AOP1160. https://projecteuclid.org/euclid.aop/1511773673


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References

  • [1] Cabello Sánchez, F. and Castillo, J. M. F. (2002). Banach space techniques underpinning a theory for nearly additive mappings. Dissertationes Math. (Rozprawy Mat.) 404 73 pp.
  • [2] Grenander, U. (1963). Probabilities on Algebraic Structures. Wiley, New York.
  • [3] Hitczenko, P. and Montgomery-Smith, S. J. (2001). Measuring the magnitude of sums of independent random variables. Ann. Probab. 29 447–466.
  • [4] Hoffmann-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52 159–186.
  • [5] Johnson, W. B. and Schechtman, G. (1989). Sums of independent random variables in rearrangement invariant function spaces. Ann. Probab. 17 789–808.
  • [6] Kahane, J.-P. (1985). Some Random Series of Functions. Cambridge Studies in Advanced Mathematics 5. Cambridge Univ. Press, London.
  • [7] Khare, A. and Rajaratnam, B. (2014). Differential calculus on the space of countable labelled graphs. Preprint. Available at arXiv:1410.6214.
  • [8] Khare, A. and Rajaratnam, B. (2015). Integration and measures on the space of countable labelled graphs. Preprint. Available at arXiv:1506.01439.
  • [9] Khare, A. and Rajaratnam, B. (2015). Probability inequalities and tail estimates for metric semigroups. Preprint. Available at arXiv:1506.02605.
  • [10] Khare, A. and Rajaratnam, B. (2016). The Khinchin–Kahane inequality and Banach space embeddings for metric groups. Preprint. Available at arXiv:1610.03037.
  • [11] Klass, M. J. and Nowicki, K. (2000). An improvement of Hoffmann–Jørgensen’s inequality. Ann. Probab. 28 851–862.
  • [12] Klee, V. L. Jr. (1952). Invariant metrics in groups (solution of a problem of Banach). Proc. Amer. Math. Soc. 3 484–487.
  • [13] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces (Isoperimetry and Processes). Springer, Berlin.
  • [14] Lovász, L. (2012). Large Networks and Graph Limits. Colloquium Publications 60. Amer. Math. Soc., Providence.
  • [15] Rudin, W. (1962). Fourier Analysis on Groups, Interscience. Wiley, New York.
  • [16] Sternberg, S. (1965). Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs.