Speech parameter generation considering global variance (GV generation) is widely acknowledged to dramatically improve the quality of synthetic speech generated by HMM-based systems. However it is slower and has higher latency than the standard speech parameter generation algorithm. In addition it is known to produce artifacts, though existing approaches to prevent artifacts are effective. We present a simple new theoretical analysis of speech parameter generation considering global variance based on Lagrange multipliers. This analysis sheds light on one source of artifacts and suggests a way to reduce their occurrence. It also suggests an approximation to exact GV generation that allows fast, low latency synthesis. In a subjective evaluation our fast approximation shows no degradation in naturalness compared to conventional GV generation.