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We study the return interval $\tau$ between price volatilities that are abovea certain threshold $q$ for 31 intraday datasets, including the Standard &Poor's 500 index and the 30 stocks that form the Dow Jones Industrial index.For different threshold $q$, the probability density function $P_q(\tau)$scales with the mean interval $\bar{\tau}$ as$P_q(\tau)={\bar{\tau}}^{-1}f(\tau/\bar{\tau})$, similar to that found in dailyvolatilities. Since the intraday records have significantly more data pointscompared to the daily records, we could probe for much higher thresholds $q$and still obtain good statistics. We find that the scaling function $f(x)$ isconsistent for all 31 intraday datasets in various time resolutions, and thefunction is well approximated by the stretched exponential, $f(x)\sim e^{-ax^\gamma}$, with $\gamma=0.38\pm 0.05$ and $a=3.9\pm 0.5$, which indicates theexistence of correlations. We analyze the conditional probability distribution$P_q(\tau|\tau_0)$ for $\tau$ following a certain interval $\tau_0$, and find$P_q(\tau|\tau_0)$ depends on $\tau_0$, which demonstrates memory in intradayreturn intervals. Also, we find that the mean conditional interval$<\tau|\tau_0>$ increases with $\tau_0$, consistent with the memory found for$P_q(\tau|\tau_0)$. Moreover, we find that return interval records have longterm correlations with correlation exponents similar to that of volatilityrecords.Comment: 19 pages, 8 figure

Scaling and memory of intraday volatility return intervals in stock markets