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O pewnym porównaniu ciągu gaussowskiego z ciągiem Rademachera
- Speaker(s)
- Stanisław Kwapień
- Affiliation
- Uniwersytet Warszawski
- Date
- Oct. 11, 2012, 12:15 p.m.
- Room
- room 3260
- Seminar
- Seminar of Probability Group
We answer a question of Weis by proving:
Theorem. If $X,Y$ are Banach spaces ($Y$-nontrivial) then $X$ has finite cotype if and only if each $\gamma$-bounded family of operators ${\cal F} \subset L(X,Y)$ is $r$-bounded.
Here: a family of operators ${\cal F} \subset L(X,Y)$ is said to be $\gamma$-bounded (resp. $r$-bounded), iff there exists a constant $c$ such that for each finite sequences of vectors $(x_n) \in X$ and operators $(T_n) \in \cal F$ it is $E||\sum T(x_n)\xi_n|| \le c E||\sum x_n\xi_n||$ where $(\xi_n)$ is i.i.d. sequence of symmetric Gaussian (resp. Rademacher) random variables.
Theorem. If $X,Y$ are Banach spaces ($Y$-nontrivial) then $X$ has finite cotype if and only if each $\gamma$-bounded family of operators ${\cal F} \subset L(X,Y)$ is $r$-bounded.
Here: a family of operators ${\cal F} \subset L(X,Y)$ is said to be $\gamma$-bounded (resp. $r$-bounded), iff there exists a constant $c$ such that for each finite sequences of vectors $(x_n) \in X$ and operators $(T_n) \in \cal F$ it is $E||\sum T(x_n)\xi_n|| \le c E||\sum x_n\xi_n||$ where $(\xi_n)$ is i.i.d. sequence of symmetric Gaussian (resp. Rademacher) random variables.
