According to this paper, Section V-C, a rational impedance matrix Z(s) is the impedance of a passive lumped system if and only if the following four conditions are satisfied:
1) each element of Z(s) is dedined and analytic in \$Re(s)>0\$;
2) \$Z(j\omega)+Z^H(j\omega)\$ is nonnegative-definite for all \$\omega\in R\$;
4) asymptotically, \$Z(s)\rightarrow As\$ in \$Re(s)>0\$, where \$A\$ is real, constant, symmetric, and nonnegative-definite.
Condition 2 ensures positive dissipated energy. Condition 3 ensures real impulse response in the time domain. My question is about the exact meaning of the fourth condition. For example, suppose I have a one-port which consists of nothing but a series of a positive resistor \$R\$ and a capacitance \$C\$. The input impedance, which is equivalent to the 1x1 impedance "matrix", is simply \$Z(s)=R+1/sC\$. So asymptotically \$Z(s) \rightarrow R\$, which is not in the form of \$As\$, for whatever \$A\$. But this simple one-port is doubtlessly passive. In this case, should we consider the 4th condition as violated? Or, does it actually mean that the asymptotic behavior of \$Z(s)\$ cannot grow faster than \$As\$?