Scaling and memory effect in volatility return interval of the Chinese stock market

We investigate the probability distribution of the volatility return intervals $\tau$ for the Chinese stock market. We rescale both the probability distribution $P_{q}(\tau)$ and the volatility return intervals $\tau$ as $P_{q}(\tau)=1/\bar{\tau} f(\tau/\bar{\tau})$ to obtain a uniform scaling curve for different threshold value $q$. The scaling curve can be well fitted by the stretched exponential function $f(x) \sim e^{-\alpha x^{\gamma}}$, which suggests memory exists in $\tau$. To demonstrate the memory effect, we investigate the conditional probability distribution $P_{q} (\tau|\tau_{0})$, the mean conditional interval $ $ and the cumulative probability distribution of the cluster size of $\tau$. The results show clear clustering effect. We further investigate the persistence probability distribution $P_{\pm}(t)$ and find that $P_{-}(t)$ decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in $\tau$. The scaling and long memory effect of $\tau$ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.