Stochastic analysis and metric measure geometry have been widely studied, but these are still thinly linked.
For example, Gromov's pyramid is an interesting object concerning the concentration of measure phenomenon and has been actively studied in geometry.
In probabilistic terms, it can be said to express the asymptotic behavior of random variables.
It is significant to investigate it probabilistically.

In this talk, we will generalize some results for Gaussian and $\ell^2$-metric to for generalized Gaussian and $\ell^\beta$-metric, and discuss a relation between our results and the ergodic theory.
In the results of generalized Cauchy-type distribution, we also give the sharp-quantitative concentration inequality around a radial function.

This talk is based on joint works with Daisuke Kazukawa(Kyushu) and Ayato Mitsuishi(Fukuoka) and another joint work with Kanji Inui(Doshisha).