Analysis of time-series data of different markets have produced evidence for several stylized facts (universal features) including heavy tails characterized by power law exponents, which provide us tantalizing hints of the dynamics underlying such complex systems. It is especially important to see how these features evolve over time after the market is created and gradually develops. The recent advent of the digital currency, Bitcoin, and its growing popularity as an asset traded between agents over the last few years, provides us with an invaluable dataset for such a study. Similar to many financial markets, Bitcoin is de-centralized and its value is not controlled by a single institution, (e.g., a central bank). Here we have analyzed high-frequency Bitcoin trading data (with a resolution of one tick, i.e., a single trading event). We show that the distribution of price fluctuation (measured in terms of logarithmic return) has a heavy tail. The exponent of the tail implies that Bitcoin fluctuations follow an inverse square law, in contrast to the inverse cubic law exhibited by most financial and commodities markets. The distribution of transaction sizes and trading volume are seen to have Levy-stable distribution. Multi-scale analysis show the presence of long term memory effects in market behavior.