Recent measurement of voltage V and current I of the electron gun of a relativistic klystron amplifier revealed that the resulting current-voltage relationship appeared to differ from the usual Child-Langmuir law (I∝V^{3/2}) especially during the initial period of voltage increase. This paper attempts to explain this deviation by examining the emission mechanism using particle-in-cell simulation. The emission area in the cathode increased stepwise as the applied voltage increased and within each step the current and voltage followed the Child-Langmuir law. The electron emission began when the voltage reached a threshold, and the perveance increased with the emission area. Furthermore, an apparent virtual cathode was formed which was larger than the cathode tip. This occurs because, above a certain voltage, the emission from the edge and the side of the cathode surface dominates the emission from the front-end surface.