The central limit theorem and the strong law of large numbers in $l_p\{X\}$-spaces, $1\le p<+\infty$

The central limit theorem is proved for independent identically distributed random elements having strong second order moments with values in a Banach space with a Shauder basis. It is shown that, if $X$ is a $G$-space, then $l_p\{X\}$, $2\le p<\infty$, is a space of the same type. The central limit theorem is also proved for the case when $1\le p\le 2$ and $X$ is a $G$-space and for $l_p\{l_s\}$-spaces where $1\le p,s\le 2$.
The strong law of large numbers in these spaces is studied.