Let $(W_i,J_i)$ be i.i.d. $\mathbb R_+\times \mathbb R$-valued random vectors and denote \[ S(n)=W_1+\dots+W_n, \qquad M(n)=\max\{J_1,\dots, J_n\} . \] We are interested in the joint convergence of $(S(n),M(n))$. More precisely, let $a_n,b_n>0$ and consider \begin{equation}\label{eq1} (a_nS(n),b_nM(n))\Longrightarrow (D,A) \end{equation} as $n\to\infty$, where $\Longrightarrow$ denotes convergence in distribution. We will address the following questions:\\ {\bf (A)} Are there necessary and sufficient conditions on the distribution of $(W_1,J_1)$ such that (1) holds?\\ {\bf (B)} Can we characterize the class of possible limit distributions in (1) ?\\ {\bf (C)} How can the possible dependence of $D$ and $A$ in (1) be characterized? Using the above results, we then analyze the limit behavior of the so-called continuous-time random maxima process defined as \[ \tilde M(t)=M(N_t) \] where \[ N_t=\max\{n\geq 0 : S(n)\leq t\} .\]