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We can't be so sure in principle, simply because there's only one realization of a process. So the notion of ergodicity has no practical value here.
I don't quite agree. We can evaluate ergodicity as a binary factor (is-no) just like any other process characteristic.
For a stationary process the ergodicity hypothesis is quite natural, for a non-stationary process it is a very strong statement to take for granted. Therefore the first step in checking for ergodicity may be to check for stationarity of a part of the time series (or some transformation of it, why not), or to identify a part where the series can be considered stationary with some certainty. Note that it is possible to do this by one realisation at a time. Further, if we were able to divide the series into ergodic sections, we can apply statistical methods on each of them without overstepping the boundaries, at least with some certainty. That seems to me to be better than nothing.
I don't quite agree. Ergodicity as a certain binary factor (is-no) we can evaluate just like any other process characteristic.
For a stationary process the ergodicity hypothesis is quite natural, for a non-stationary process it is a very strong statement to be taken on faith. Therefore the first step in testing for ergodicity may be to check for stationarity of some part of the time series (or some transformation of it, why not), or to identify a part where the series can be considered stationary with some certainty. Note that it is possible to do this by one realisation at a time. Further, if we were able to divide the series into ergodic sections, we can apply statistical methods on each of them without overstepping the boundaries, at least with some certainty. That seems better than nothing to me.
As said above, the exploitation of the hypothesis consists in 'trusting' various kinds of time averages on ergodic plots and 'distrusting' on non-ergodic ones... in a kind of generalised sense, so to speak.
More specifically, we can give the following example of incredulity: if I
(a) Received a signal for input using some kind of time averages and the hypothesis that they can replace the deterministic component, i.e. the ensemble average,
b) and at the same time I have information that the process was essentially non-stationary/non-ergodic in the analysis section,
then I do not trust such a signal.
It is not all that straightforward. The article from the handbook applies only to differentiable processes, while stochastic processes, i.e. those with a random component, do not formally belong to such processes: the limit dS/dt does not exist, hence there is no derivative. As stated above, the price can "wiggle" at any small interval of time, and we cannot get inside this interval for purely technical reasons.
That's why I think the question has a non-trivial meaning.
Why is there no limit? A tick is a limit. So we divide the value of a tick (change per tick) at the moment of its occurrence by the time since the previous tick. Dimension is point/second. There is no more limit))
Whether to average or not depends on the specific task and can be deduced by testing
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D. N. Zubarev.
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Very important and very strict (!!!) conditions of applicability of the ergodicity hypothesis are (1) closure of the system and (2) equilibrium of the system.
Neither of these conditions is satisfied by the market.
1) It is an open system.
2) It is a highly non-equilibrium system.
Methods for studying open non-equilibrium systems do not use the ergodicity hypothesis. (And they do not need such a hypothesis.)
Very important and very rigid (!!!) conditions of applicability of the ergodicity hypothesis are (1) closedness of the system
No. The paper describes the ergodicity condition for a closed system, not closedness as a condition. Therefore
1) The market is an open system.
is not an obstacle to ergodicity. The other is,
(2) Equilibrium of the system.
This condition is essential, but the assertion
2) Market is a highly non-equilibrium system.
is not always true. There are areas of equilibrium, or areas that can be reduced to equilibrium by a simple transformation (e.g. subtracting demolition, accounting for seasonality, etc.). This is exactly what I was talking about.
Otherwise, of
Methods for studying open non-equilibrium systems do not use the ergodicity hypothesis. (and do not need such a hypothesis)
follows the impossibility to apply the apparatus of matstatistics to the market in principle, as it substantially relies on the ergodicity hypothesis.
By the way, statistical physics needed the ergodicity hypothesis in order to justify the application of mathematical statistics, without this hypothesis all statistical calculations at least for gas, at least for the market are tantamount to shamanism.
Just in case, a counter-example.
A stationary random process is fed to the input of a linear differential filter. The output is also a stationary process.
We have:
1) the system is open
2) ergodicity hypothesis is satisfied, as all time averages are obviously equal to the population mean - expectation, variance, etc., if only they exist.
Just in case, here's a counterexample.
A stationary random process is fed to the input of a linear filter - a differentiating link. The output is also a stationary process.
We have:
1) the system is open
2) ergodicity hypothesis is satisfied, as all time averages are obviously equal to the population mean - expectation, variance, etc., if only they exist.
This is a bad counterexample. It's very limited.
As an example, consider a more appropriate model for our case: Some finite volume of a compressible viscous fluid, with a bounded surface, and in motion -- a process accompanied by mechanical work, heat exchange with the external environment, conversion of mechanical energy into heat.
The calculations are more complicated, but much more interesting.
This is a bad counterexample. Very limited.
As an example, consider a more appropriate model for our case: Some finite volume of a compressible viscous fluid, with a bounded surface, and in motion -- a process accompanied by mechanical work, heat exchange with the external environment, conversion of mechanical energy into heat.
The calculations are more complicated, but much more interesting.
The question is: "Can you even describe the quadratic trinomial?
The answer is, 'No, I can't even imagine it'.