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### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 6:58 pm
QuoteOriginally posted by: vixenQuoteIf you average over enough time to remove the time dependency, I agree with most. However, if you don't use the QM average the world approach, Cauchy dynamics is 100% deterministic scattering.We are not talking about time dependence or dynamics in the context of the CLT!Mean of a large number of iid rv from a Cauchy distribution follows a Cauchy distribution. Do you agree or not?I've always found it weird how you could not talk about time dependence. What precisely is a "large number of iid rv variables" and where do they come from? If i'm sampling from some system I can only receive a finite and relatively small number of samples before you know, time actually flows forward an appreciable amount, so does the CLT apply to finance? If so, on what timescale?

### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 7:14 pm
Quote I've always found it weird how you could not talk about time dependence. What precisely is a "large number of iid rv variables" and where do they come from? If i'm sampling from some system I can only receive a finite and relatively small number of samples before you know, time actually flows forward an appreciable amount, so does the CLT apply to finance? If so, on what timescale? Sure. I appreciate your point. I understand that there are issues with how accurately you can model reality. And how large is a large number and all that. I was just making sure that N is not relying on a limited version of the CLT ( ie Gaussian ) to make his case about randomness.My mistake: I think that in the case of the standard Cauchy distribution you don't even need a large number! Any finite number will do.As far as applicability to finance, you can make a far stronger case for the distribution of returns to be Levy-Stable than Gaussian!So if the distribution of daily returns has the same functional form as the distribution of monthly returns, then, since one is the sum of the other, the distribution has to be alpha-stable ( of which Gaussian is a special case ) and follow the generalized CLT.

### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 7:41 pm
QuoteOriginally posted by: crowlogicIf i'm sampling from some system I can only receive a finite and relatively small number of samples before you know, time actually flows forward an appreciable amountOnly one sample, in fact, because, in a time series, each sample is separated by a finite time difference.

### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 7:41 pm
QuoteOriginally posted by: vixenQuoteIf you average over enough time to remove the time dependency, I agree with most. However, if you don't use the QM average the world approach, Cauchy dynamics is 100% deterministic scattering.We are not talking about time dependence or dynamics in the context of the CLT!Mean of a large number of iid rv from a Cauchy distribution follows a Cauchy distribution. Do you agree or not?We are not taking in any way about the CLT, means, probability or distributions. The CLT does not apply here. Samples of Cauchy 'distributions' are never independent. Adding samples (at t1...tn) together will produce a sum that has cauchy dynamics or has no dynamics at all.

### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 7:45 pm
QuoteOriginally posted by: FermionQuoteOriginally posted by: crowlogicIf i'm sampling from some system I can only receive a finite and relatively small number of samples before you know, time actually flows forward an appreciable amountOnly one sample, in fact, because, in a time series, each sample is separated by a finite time difference.Well yes, but sometimes stuff flows in so fast that the resolution of the clock on a digital computer doesn't even have time to change.

### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 7:53 pm
QuoteOriginally posted by: NQuoteOriginally posted by: vixenQuoteIf you average over enough time to remove the time dependency, I agree with most. However, if you don't use the QM average the world approach, Cauchy dynamics is 100% deterministic scattering.We are not talking about time dependence or dynamics in the context of the CLT!Mean of a large number of iid rv from a Cauchy distribution follows a Cauchy distribution. Do you agree or not?We are not taking in any way about the CLT, means, probability or distributions. The CLT does not apply here. Samples of Cauchy 'distributions' are never independent. Adding samples (at t1...tn) together will produce a sum that has cauchy dynamics or has no dynamics at all.So tell us N, when does CLT apply? only when you want to prove a point? And please do tell us why are INDEPENDENT samples from a cauchy distn are not independent?

### Brownian motion, does this phrase make sense?

Posted: May 9th, 2007, 8:03 pm
QuoteOriginally posted by: NDistribution = PDF(x).Stationary = PDF is not a function of t.If we have PDF(x,t) that's not a distribution.If we are taking about finance, instead of arguing pure math for the sake of it, then this is a Humpty-Dumpty argument. ("When I use a word it means just what I want it to mean"). I have never had much interest in pure math, but I'm very good at finding simple math to describe complex real-life problems. I rarely need much more than undergraduate calculus and algebra. So forgive me if you find it tiresome to bring this discussion back to earth; I couldn't join you up in the clouds even if I tried.A variable transformation y = Y(x,t) converts a stationary PDF into a non-stationary one or vice versa. It's the stationarity of the model used for the time series (i.e. getting the time-dependence of the PDF right -- or as right as does the job) that matters, not stationarity of the PDF itself. Alternatively and equivalently, it's about finding the variable for which the empirical PDF is effectively stationary.To bandy terms like stationary about without specifying context is just as silly as using words like random and deterministic without specifying variables. So when you're ready, N, to specify which variables are deterministic or have a stationary PDF, I'm all ears.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 4:55 pm
QuoteOriginally posted by: crowlogicQuoteOriginally posted by: FermionQuoteOriginally posted by: crowlogicIf i'm sampling from some system I can only receive a finite and relatively small number of samples before you know, time actually flows forward an appreciable amountOnly one sample, in fact, because, in a time series, each sample is separated by a finite time difference.Well yes, but sometimes stuff flows in so fast that the resolution of the clock on a digital computer doesn't even have time to change.I would probably prefer (a necessity if the market is efficient) to calibrate my digital clock to the samples. Each event is one tick on the clock.If successive events are treated as independent samples of the same phenomenon then the time scale (or number of events/ticks) over which this is a reasonable assumption is the time scale over which market memory dominates market efficiency.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 5:06 pm
QuoteOriginally posted by: FermionQuoteOriginally posted by: crowlogicQuoteOriginally posted by: FermionQuoteOriginally posted by: crowlogicIf i'm sampling from some system I can only receive a finite and relatively small number of samples before you know, time actually flows forward an appreciable amountOnly one sample, in fact, because, in a time series, each sample is separated by a finite time difference.Well yes, but sometimes stuff flows in so fast that the resolution of the clock on a digital computer doesn't even have time to change.I would probably prefer (a necessity if the market is efficient) to calibrate my digital clock to the samples. Each event is one tick on the clock.If successive events are treated as independent samples of the same phenomenon then the time scale (or number of events/ticks) over which this is a reasonable assumption is the time scale over which market memory dominates market efficiency.In reality you have no control over which samples you receive(you can however choose to disregard things) and they are not evenly spaced in time and are so wildly discontinuous that you cannot interpolate. So really even a 1 dimensional time series is actually 2 dimensional cause the time-delta is the 2nd dimension.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 9:26 pm
QuoteOriginally posted by: outrunIn my opinion returns are noise driven chaotic systems. There are event that are unpredictable (giving N a trading account would be such a event generator), but sometimes you'll see deterministic behavior. E.g. after an extreme move (takeover bid) you'll see oscillating movement with a frequency of hours that always look the same. I'm a big fan of nonparametric methods. Why pick a model a-priori & fit it to the data? It will never contain more info that the data. Regarding Stable distributions: they allow you to do something with the observation that the hurst exponent <> 0.5, but there is no reason to postulate that the distribution should be timescale invariant. Furthermore, you'll need to model stochastic behavior with transition probabilities, not just distributions.I agree with you that there is no need to postulate that the distribution should be timescale invariant.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 9:35 pm
if u return to a physics roots of BM then timing scales are not an issue. dust particle's movement is continuous, it's only that it gets bombarded by smaller molecules of the host liquid. when u decrease the sampling time u rach to the point when it's comparable to average delay between hits, which is a function of liquid density and temperature and size of the dust particle. at this point nothing happens to dust, it floats peacefully.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 9:56 pm
QuoteOriginally posted by: jawabeanif u return to a physics roots of BM then timing scales are not an issue. dust particle's movement is continuous, it's only that it gets bombarded by smaller molecules of the host liquid. when u decrease the sampling time u rach to the point when it's comparable to average delay between hits, which is a function of liquid density and temperature and size of the dust particle. at this point nothing happens to dust, it floats peacefully.Yup, assuming things at in equilibrium or assuming some sort of dynamic equilibrium, they get more complex if the properties of the medium itself change with time, as they do of course.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 10:46 pm
QuoteOriginally posted by: crowlogicYup, assuming things at in equilibrium or assuming some sort of dynamic equilibrium, they get more complex if the properties of the medium itself change with time, as they do of course.equilibrium means that those properties change "slowly" compared to characteristic times of your process. if the temperature of liquid changes a bit, it's still Ok with BM of a dust particle. however, if there's turbulence in liquid, then it's all screwed up.but the point was that the fact that you can't limit time interval to zero is not a big issue.

### Brownian motion, does this phrase make sense?

Posted: May 10th, 2007, 10:49 pm
QuoteOriginally posted by: jawabeanQuoteOriginally posted by: crowlogicYup, assuming things at in equilibrium or assuming some sort of dynamic equilibrium, they get more complex if the properties of the medium itself change with time, as they do of course.equilibrium means that those properties change "slowly" compared to characteristic times of your process. if the temperature of liquid changes a bit, it's still Ok with BM of a dust particle. however, if there's turbulence in liquid, then it's all screwed up.but the point was that the fact that you can't limit time interval to zero is not a big issue.Yup. So, are stock prices turbulent or follow a BM? I think the former.