The traditional efficiency market hypothesis supposes that the fluctuations of the stock prices are random, and stock price is unable to be predicted. But in recent years papers point out that the fluctuations of the stock prices are not totally random, the fluctuations of the stock price have long term memory characteristics. Therefore, trying to find out the regularity of the market price becomes a new subject for research. This paper attempts to use the fractional market hypothesis to analyze stock market, which divided samples into two types which are the developed markets ¡]Japan , U.S.A. , Australia, and South Africa¡^and mergering markets ¡]Korea , Taiwan , China¡]Shanghai¡^ , and Jordan¡^. The sample period is from January of 1997 to December of 2006. And using the regarding countries¡¦ main returns of daily stock price index. By using R/S analysis to estimate each country¡¦ Hurst coefficients, this paper studies the aperiodic cycle in each country. It also wants to see whether the degree of maturity affects the different result or not. The empirical results show that the stock indeies in the developed markets have shorter aperiodic cycle than in the mergering markets. U.S.A., Australia ,and South African markets the aperiodic cycles are 138 days , 126 days ,and 152 days respectively. Taiwan and Shanghai markets the aperiodic cycles are 208 days and 202 days respectively. Japan, Korea , Jordanian markets in this sample period have not found aperiodic cycles.