Let \(\{X, X_{n, k}: 1\leq k\leq n, n\geq 1\}\) be an array of i.i.d. \(B\)-valued random variables, where \((B, |\cdot|)\) is real separable Banach space. Set \(S(n)= X_{n, 1}+ X_{n, 2}+\cdots+ X_{n, n}\), \(n\geq 1\), and \(\text{Log } t= \log\max\{e, t\}\), \(t\in \mathbb{R}\). The main result states that \(\{S(n)/\sqrt{2n\text{ Log } n}, n\geq 1\}\) is conditionally compact in \(B\) with probability one if, and only if \(EX= 0\), \(E(|X|^4(\text{Log}|X|)^{-2})< \infty\) and \(\{S(n)/\sqrt{2n\text{ Log } n}\}\) converges to \(0\) in probability. Certain other aspects of the asymptotic behavior of \(\{S(n)/\sqrt{2n\text{ Log } n}\}\) are studied as well as the case of row-wise independent \(B\)-valued arrays.