Random Flow Theory and FOREX - page 68
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In eagle, if heads are 1, tails -1 then MO=0, D(X)=((0-1)^2+(0+1)^2)/2=1
Conant dispersion and constant MO. Why non-stationary?
Even if we take a cumulative sum over any fixed number of shots (e.g. 100), the distribution would be normal with MO=0 and a fixed, easily calculated variance.
That is exactly why it is non-stationary, because for cumulative sum it will be different depending on how many shots you take into account, i.e. the second point depends on time (in this case on number of shots). The definition of stationarity is that the first and second moments are NOT time dependent.
So the generating process - binomial - will have variance equal to 1 always, regardless of how many tosses there are. It is a stationary process.
To go further, the cumulative sum - random walk - "remembers" all previous results, it has a long memory. Binomial remembers nothing at all about past rolls, i.e. its memory is so short that it is zero.
You're not getting it right again, brother. Or do you really think I want something from you? :-)
I'm used to finding confirmation or refutation on all questions that interest me.
And in this case I just wanted you to give me a receipt for your hollowness. Which you did. Congratulations.
"Meli Imelia, your week" - Mr "stationary random rambling".
This is exactly why it is non-stationary, because for the cumulative sum variance will be different depending on how many throws you take into account, i.e. the second momentum depends on time (in this case the number of throws). The definition of stationarity is that the first and second moments are NOT time dependent.
So the generating process - binomial - will have variance equal to 1 always, regardless of how many tosses there are. It is a stationary process.
To go further, the cumulative sum - random walk - "remembers" all previous results, it has a long memory. Binomial remembers nothing at all about past rolls, i.e. it has such a short memory that it is zero.
Timbo, the cumulative sum has, as you deign to put it, a DIFFERENT (UNLIMITED) DISPERSION. You don't even have to be a mathematician to know this "paradox" - you just had to read Schwager's books on trading.
Look, colleagues, personally, I am tired of clearing up your clutter here. There are more interesting things to do in life. As soon as there is a sensible conversation between responsible people, I will come back to this thread.
Goodbye.
Here, instead of me, is the link, it says it all:
http://www.wikipedia.org
This is exactly why it is non-stationary, because for the cumulative sum variance will be different depending on how many throws you take into account, i.e. the second momentum depends on time (in this case the number of throws). The definition of stationarity is that the first and second moments are NOT time dependent.
So the generating process - binomial - will have variance equal to 1 always, regardless of how many tosses there are. It is a stationary process.
To go further, the cumulative sum - random walk - "remembers" all previous results, it has a long memory. Binomial remembers nothing at all about past rolls, i.e. it has such a short memory that it is zero.
Sorry, but we apparently have different concepts of "stationary distribution". Time-independent means not changing over time, not depending on the amount of time for the counts. In the coin example above, the variance for samples with a sample rate of 1 flip does not change over time. It is constant both at the beginning and after a thousand tosses. I.e. the increment is a stationary process. The cumulative sum is also a stationary series. You can calculate the variance in the same way, and it does not change over time. Though it is possible to break it down differently, for example as I wrote in the series of shots (by 100 for example), and the increments will still be a stationary series (and the cumulative sum too). That's why a dozen pages earlier I wrote that it's not the process that is stationary or non-stationary, but the breakdown into a series of observations.
Infinite variance is indeed a property of a non-stationary process. For example, the increments will not be Gaussian distributed, but with "thick tails" and a couple of other differences. At first sight the differences are not principal but they change the situation cardinally, especially for the risk accounting.
Either the cumulative sum has infinite variance, in which case it cannot be a stationary process, or the sum is stationary, in which case its variance is a constant (finite) value for any series length.
I suggest not using the word "increment" at all for the time being. We estimate the sum of these increments, i.e. the random walk, and we'll discuss what it came from later.
Can you provide references to "your" definition of stationarity. Not from memory, but a citation to a decent source. Wikipedia is quite a decent source when it comes to statistics.
Young man, I've been waiting for you to correct your mistake, which has been politely pointed out to you, but you are not itching and don't think to correct it.
I do not claim to have an academic degree, I am allowed. If you have such baggage of knowledge, this forum is not for you. The task was to show that the distribution of three sigmas is easy to get, there are too many fat-tailed animals.
Either the cumulative sum has infinite variance, in which case it cannot be a stationary process, or the sum is stationary, in which case its variance is a constant (finite) value for any series length.
I suggest not using the word "increment" at all for the time being. We estimate the sum of these increments, i.e. the random walk, and we'll discuss what it came from later.
Can you provide references to "your" definition of stationarity. Not from memory, but a citation to a decent source. Wikipedia is a pretty decent source when it comes to statistics.
The concepts of variance, stationarity etc are defined for a series. Which series are you considering? It all depends on it.
Take a coin and its cumulative amount. This is the series. It is equal to the previous value + increment. Since the MO of the increment is zero, the MO of the next term in the series will be equal to the previous value, and the variance will be equal to the variance of the increment (one). Thus the variance does not change, and the MO does not carry a random component and is unambiguously determined at any point in time. We have this initial series and then we can make another series out of it, for example by dividing it into series of fixed length. This new series will be stationary. Its MO will be equal to finite value of cumulative sum of previous term, and dispersion can be easily calculated (increments will be distributed normally).
The original series could have been split differently: not by a fixed length, but by a variable, for example. In this case the new series will be non-stationary - its variance will vary. It all depends on the partitioning of the original series. For example, if we take EUR watch (time interval of 1 hour), then its distribution will be non-stationary, although it does not exclude the possibility of other sampling where the distribution will be stationary. And not necessarily in time.
Stationarity is a property of a probabilistic process to remain constant over time. AlexEro gave a more detailed definition on 'Random Flow Theory and FOREX'
and further to distributions are invariant with respect to time shifts. That is, it remains unchanged with time shifts.
I asked you not to use the word increment. By doing any partitioning, you are again talking about increments, and the question is about the cumulative sum. The process is like this. Random wandering. Whether it is stationary as some comrades here claim or not as I claim.
Its MO will be equal to the final value of the cumulative sum of the previous term, and the variance is also easy to calculate (the increments will be distributed normally).
Ease of calculation is not a criterion for stationarity.
Example with a coin (1,-1) - cumulative sum: if a series of one flip, then the variance of the cumulative sum is 1; if two flips, then it is 2, if three flips, then almost 4, and so on. That is, the variance depends on the length of the series.
Now compare it with the process of just flipping a coin: no matter how many times you flip it, the variance is still 1, i.e. it does not depend on the length of the series.
I asked you not to use the word increment. By doing any partitioning, you are again talking about increments, and the question is about the cumulative sum. The process is this. Random wandering. Is it stationary as some comrades here claim or not as I claim.
I was actually talking about the cumulative sum. Just broke it down to calculate variance and MO. Cumm. sum is equal to the previous value + 1/-1(eagle/tree), right?
What series are you talking about?
For example, a series of eagle/rarek drop: ORROROR=+1+1-1-1-1+1+1+1-1-1-1
Cumulative sum: 0;1;2;1;0;-1;0;-1;0;1;-1;-2
The cumulative sum series is stationary. The variance=1 for each term of the series. If you break it up into series of variable length, the new series is not stationary. What you probably mean is that if you calculate MO and variance for a series of the entire length of the series (14 values), then if you then continue the series and calculate variance for more values (100 for example), it will be larger and will increase with the number of members in the series. I don't argue with that and wrote about series of variable length. Such series will be non-stationary. In short, everything depends on the partitioning of the initial series, but the initial series is stationary.