Some Results on the Central Limit Theorem for Subsequences in Banach Spaces

Let $\{X,X_n;n\ge 1\}$ be a sequence of i.i.d. $\mathbf{B}$-valued random variables and set $S_n={\sum }_{i=1}^n{X}_i,n\ge 1$. This note is devoted to study the classical central limitr theorem for subsequences of sums of i.i.d. $\mathbf{B}$-valued random variables. We show that, under the assumption that $\mathbf{B}$ is of cotype $2$ space, ${\left(\frac{S_n}{\sqrt{n}}\right)}_{n\ge 1}$ converges weakly if and only if ${\left(\frac{S_{m_n}}{\sqrt{m_n}}\right)}_{n\ge 1}$ converges weakly for a subsequence $\{m_n;n\ge 1\}$ of positive integers. We conjecture that this result is false if $\mathbf{B}$ is not of cotype $2$ space. In addition, we show that, if ${\left(\frac{S_{m_n}}{\sqrt{m_n}}\right)}_{n\ge 1}$ converges weakly for a subsequence $\{m_n;n\ge 1\}$ of positive integers. and ${\left(\frac{S_n}{\sqrt{n}}\right)}_{n\ge 1}$ does not converge weakly, then ${\displaystyle {\left(S_n\mathrm{/}{a}_n\right)}_{n\ge 1}}$ does not converge weakly to a non-degenerate probability measure for any sequence $\{a_n;n\ge 1\}$ of positive real numbers.