Let $Z_i = (X_i, Y_i), i \geqq 1$, be independent two-dimensional random variables, defined on a probability triple $(\Omega, \mathscr{A}, P)$, such that $E(X_i) = E(Y_i) = E(X_i Y_i) = 0, E(X_i^2) < \infty, E(Y_i^2) < \infty$ for all $i$. The purpose of this paper is to investigate the limit points of $\{(S_n(\omega)/L(n), T_n(\omega)/M(n)), n = 1,2,\cdots\}$, where $\omega \in \Omega, S_n = \sum^n_{i=1} X_i, T_n = \sum^n_{i=1} Y_i, L(n) = \lbrack 2E(S_n^2) \log \log E(S_n^2) \rbrack^{\frac{1}{2}}, M(n) = \lbrack 2E(T_n^2) \log \log E(T_n^2) \rbrack^{\frac{1}{2}}$. The author will show the limit sets are the closed unit disk almost surely under some general conditions. An example with all limit points lying on the two axes with probability one will be constructed.