I think it is important to consider what 'stationary' and 'non-stationary' actually mean, and fundamentally refers to dependence on time, but can be interpreted in different ways. One way of defining a stationary process is to say that it is time-independent, which means that the average value of the measurements is a constant. However, right away you see that this may be problematic in the sense that if the average is not constant but changes slowly over time, you will only be able to detect this if the time scale over which you make your measurements is significantly longer than the timescale over which the average value changes. On the other hand, the process could be fluctuating quite a lot on short time scales, but retain a constant average over the long term. So, there is some ambiguity here that one needs to be aware of when considering stationarity.

Looking at the histogram of all your measurements gives you another means to evaluate stationarity: if it is a symmetric distribution, then you can be pretty confident that the process is stationary. But if it is not, then the process may or may not be stationary. For example, you could have a non-symmetric Poisson distribution if the number of measurements in each time interval is small, but the process could well be stationary. So this brings us to the concept of the underlying distribution being stationary.

The simplest way to generate a stochastic time-independent process is to draw deltaTs (the time between events that need to be cumulatively summed to give arrival times) from an exponential distribution. Plotting the number of events as a function of time yields a flat time series with a constant average given by 1/mean of the exponential distribution. But imagine, instead of drawing each event from the same exponential, first drawing the mean of the exponential first from a very narrow gaussian, and then drawing the next deltaT from the exponential whose mean is that gaussian random number. The resultant time series will most likely appear to have a constant average but will inevitably have larger fluctuations than the first time series. Going further, image using an ever-wider gaussian distribution from which you draw the mean of the exponential from which you draw the deltaT. This will result in ever-increasing fluctuations, and possibly the appearance of structures in the time series, but maybe not. Nonetheless, over the entire time series, the average value will appear to be constant but with a larger variance. So, it that process stationary? And if you were to make measurements of such a process instead of simulating it, how can you tell if it is or not? The use of periodograms and other timing analysis methods would probably come in handy.

In any case, I just wanted to illustrate that it is important to consider things carefully, and take the time to look into the problem at hand. And then, clearly define and express what we are measuring, what we are assuming and how we are caring out our analysis. Hope this is helpful.

Take some object, freely floating on the water (let's say a boat without rows). Consider two types of water environments: a lake and a river. Let's measure object's distance from the centre of the lake, and, in river's case, the distance from the start of the river (influx). The distribution of distances through time would be stationary in lake an non-stationary in river. That's just my guess. =). I'd rather call distribution 'symmetric', if it's resulting from stationary time series. Since all I've heard about 'stationarity' vs 'non-stationarity' was in time series context, where stationary series are fluctuating more predictably and not trending away from the plot boundaries. Main force resulting in those distributions would be wind in lake and stream in river and I imagine boat in a river could be stuck more often and more unpredictably and also distance would never decrease, hence a clear trend. After de-trending, there are chances you would achieve stationarity, especially if the river is not too curvy and boat don't get a chance to get stuck randomly.

Your question falls in spatial analysis. From hydrological point of view if a lakes, ponds or groundwater are polluted at different level of certain scates of pollutants and no change is observed in that level of pollution over time, it is stationary distribution of pollution. in the same way in normal conditions of precipitation soils or hydrological groups of soils are observed to have specific rate of infiltration. On the other hand, if there is landfill having pollutants, through leachet these pollutants go down to groundwater and move with groundwater following law of groundwater movement, the pollution level over time will not same everywhere. In the same manner some biodegradable pollution as excreta, pulp or waste from sugar industry will cause oxygen sag, but after sometime the dissolved leve of oxygen will recover as the effluent move away with the current of stream.

I think it is important to consider what 'stationary' and 'non-stationary' actually mean, and fundamentally refers to dependence on time, but can be interpreted in different ways. One way of defining a stationary process is to say that it is time-independent, which means that the average value of the measurements is a constant. However, right away you see that this may be problematic in the sense that if the average is not constant but changes slowly over time, you will only be able to detect this if the time scale over which you make your measurements is significantly longer than the timescale over which the average value changes. On the other hand, the process could be fluctuating quite a lot on short time scales, but retain a constant average over the long term. So, there is some ambiguity here that one needs to be aware of when considering stationarity.

Looking at the histogram of all your measurements gives you another means to evaluate stationarity: if it is a symmetric distribution, then you can be pretty confident that the process is stationary. But if it is not, then the process may or may not be stationary. For example, you could have a non-symmetric Poisson distribution if the number of measurements in each time interval is small, but the process could well be stationary. So this brings us to the concept of the underlying distribution being stationary.

The simplest way to generate a stochastic time-independent process is to draw deltaTs (the time between events that need to be cumulatively summed to give arrival times) from an exponential distribution. Plotting the number of events as a function of time yields a flat time series with a constant average given by 1/mean of the exponential distribution. But imagine, instead of drawing each event from the same exponential, first drawing the mean of the exponential first from a very narrow gaussian, and then drawing the next deltaT from the exponential whose mean is that gaussian random number. The resultant time series will most likely appear to have a constant average but will inevitably have larger fluctuations than the first time series. Going further, image using an ever-wider gaussian distribution from which you draw the mean of the exponential from which you draw the deltaT. This will result in ever-increasing fluctuations, and possibly the appearance of structures in the time series, but maybe not. Nonetheless, over the entire time series, the average value will appear to be constant but with a larger variance. So, it that process stationary? And if you were to make measurements of such a process instead of simulating it, how can you tell if it is or not? The use of periodograms and other timing analysis methods would probably come in handy.

In any case, I just wanted to illustrate that it is important to consider things carefully, and take the time to look into the problem at hand. And then, clearly define and express what we are measuring, what we are assuming and how we are caring out our analysis. Hope this is helpful.

In fact we talk about the stationarity of a series, not about the stationarity of a distribution.

We can talk about the strictly stationarity, or weakly stationarity, according the attached file (the book has the title as in the name of file, and the authors are Brockwell and Davis).

Definition 1: The time series X[1],...,X[n] is strictly stationary if the cumulative distribution function of X[t] does not depend on t between 1 and n, and the cumulative distribution function of the bivariate random variable (X[t],X[t+k]) with t and t+k between 1 and n depends only on the lag k.

Definition 2: The time series X[1],...,X[n] is weakly stationary if the expectation of X[t] with t between 1 and n, E(X[t]), does not depend on t, and the covariance between X[t] and X[t+k] with t and t+k between 1 and n depends only on the lag k.

Remark: If k=0 we have, of course, the variance of X[t].

What is the current status of generalised distribution models suitable for non-stationary processes? For example, I remember WVD from my signal processing days in late 1980s:

http://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution

Thank you all for your replies.

http://iisc-wree.com/forum/index.php?topic=161.new#new

I found this link helpful for my understanding of non stationary distribution.