## The Annals of Probability

### An improvement of Hoffmann-Jørgensen’s inequality

#### Abstract

Let $B$ be a Banach space and $\mathscr{F}$ any family of bounded linear functionals on $B$ of norm at most one. For $x\inB\set \|x\| = \sup_{\Lambda \in \mathscr{F}} \Lambda(x)(\|\cdot\|$ is at least a seminorm on $B$). We give probability estimates for the tail probability of $S_n^*= \max_{1 \leq k \leq n}\|\sum_{j=1}^{k} X_j\|$ where $\{X_i\}_{i=1}^{n}$ are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that $S_n^*$ exceeds a threshold defined in terms of $\sum_{j=1}^k Y^{(j)}$, where $Y^{(r)}$ denotes the $r$th largest term of $\{\|X_i\|\}_{i=1}^n$. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as $E\Phi(\|S_n|\|)$ and $E\Phi(S_n^*)$ follow (for any fixed $1 \leq n \leq \infty)$. Included in this paper are uniform $\mathscr{L}^p$ bounds of $S_n^*$ which are within a factor of 4 for all $p \geq 1$ and within a factor of 2 in the limit as $p \to \infty$.

#### Article information

Source
Ann. Probab., Volume 28, Number 2 (2000), 851-862.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160262

Digital Object Identifier
doi:10.1214/aop/1019160262

Mathematical Reviews number (MathSciNet)
MR1782275

Zentralblatt MATH identifier
1044.60010

#### Citation

Klass, Michael J.; Nowicki, Krzysztof. An improvement of Hoffmann-Jørgensen’s inequality. Ann. Probab. 28 (2000), no. 2, 851--862. doi:10.1214/aop/1019160262. https://projecteuclid.org/euclid.aop/1019160262

#### References

• de Acosta,A. (1980). Strong exponential integrability of sums of independent B-valued random vectors. Probab. Math. Statist. 1 133-150.
• Hoffmann-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52 159-186.
• Klass,M. J. (1981). A method of approximating expectations of functions of sum of independent random variables. Ann. Probab. 9 413-428.
• Latala,R. (1997). Estimation of moments of sums of independent real random variables. Ann. Probab. 25 1502-1513.
• Montgomery-Smith,S. (1990). Distributions of Rademacher sums. Proc. Amer. Math. Soc. 109 517-522.
• Talagrand,M. (1988). An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc. 104 905-909.
• Talagrand,M. (1989). Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab. 17 1546-1570.
• Yurinskii,V. V. (1974). Exponential bounds for large deviations. Theory Probab. Appl. 19 154-155.