In recent decades, the carbon emission allowances are bought and sold not only by carbon emitters but also by investors. We analyzed the data of price of the European Union allowances (EUA) futures in European Climate Exchange (ECX) market. The symmetric and power-law probability density function of return time series was displayed. We find that two type correlaitons exist, short-range correlations in price changes (return), while long-range correlations in the absolute of price changes (volatility). Further, we applied detrended fluctuation analysis (DFA) approach to deduce the Hurst exponent which may quntify long-range autocorrelations. We observed long-range power-law autocorrelations in the volatility which usually be used as a risk factor. Like the behavior of stock market, the correlation of EUA volatility decay much more slowly than the autocorrelation of returns. We find that the significant cross correlations exist between return time series of EUA and many other returns. These cross correlations exist in a wide range of fields, including stock markets, energy concerned commodities futures, and financial futures. The significant cross-correlations between energy concerned futures and EUA indicate the physical relationship between carbon emission and energy production process.