How to test if a time series becomes stationary? I am new to time-series analysis, but I am dealing with a non-seasonal time series data now.
The plot of the data looks like the following, so there might be huge variance at the beginning, and eventually it becomes stable, we believe the curve would finally become stationary as time goes by. 

The problem is, how could I test if it becomes stationary eventually? or could I detect a certain period that is locally stationary in the data?
 A: Time series data is commonly made stationary through differencing, i.e. computing the difference between consecutive observations of the series. Nearly all time series data does not need to be differenced more than two times in order to achieve stationarity and there are a few ways to detect stationarity in the data. 
1) You can use the autocorrelation function to determine if your data is stationary. The 'stats' package in R has a function titled acf() that will plot this for you; if the data is stationary, your plot should lie within the 95% limits imposed.
2) Compute the Ljung-Box Q statistic AFTER you fit an ARIMA model to the series. The test must be computed after as it analyzes the residuals of the model. The Box.test() function in R, from the same 'stats' package, will compute this; if your p > .05, you fail to reject the null hypothesis that the change in the data from observation to observation is inherently random.
3) Simply plot the time series each time after it has been differenced. If the series looks like random noise around the mean of the data with no discernible shape, you can probably infer that the data has achieved stationarity. 
There is a great website that explains not only these concepts but ARIMA forecasting in general for time series data. It's certainly not the only way you can do it, but I am a fan. The link is below. Hope it helps!
https://www.otexts.org/fpp/8/1
