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On Orlicz spaces satisfying the Hoffmann-Jørgensen inequality

Dominik Kutek
Uniwersytet Warszawski
May 11, 2023, 12:15 p.m.
room 3160
Seminar of Probability Group

We say that an Orlicz function \Psi satisfies the H-J inequality, if \| \sum_{k=1}^n X_k \|_{\Psi} \le C_{\Psi} ( \| \sum_{k=1}^n X_k \|_{L_1} + \| \max_{k \le n}\|X_k\| \|_{\Psi}) for any collection \{X_k\}_{k \le n} of zero mean and independent random variables taking values in any separable Banach space (F,\|.\|). Inequality of this type firstly appeared in the work of Hoffmann-Jørgensen in the case of L_p norms, and was later investigated by Talagrand in the case of \Psi(x) = \exp(|x|^a)-1, a \in (0,1] and sup-exponential Orlicz functions. Recently, it has been proved by Chamakh-Gobet-Liu that the functions of type \Psi(x) = \exp( \ln^b(1+x)) - 1, b>1 also satisfy the H-J inequality.

The main part of the talk will be devoted to the characterisation of all Orlicz functions \Psi satisfying the forementioned H-J inequality, where the necessary and sufficient condition is obtained in terms of a simple inequality on \Psi. Later, we shall discuss some applications concerning concentration inequalities and L_{\Psi}(F) boundedness of sums of independent random variables.