Theorem on the presence of memory in random sequences - page 28

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Alexey Burnakov:

I will think about it. I myself have been looking for dependencies specifically on forex market returns using the mutual information method and continue to do so. It is there.

But here, as I understand it, we are talking about an arbitrary series.

Not arbitrary, but random.

Some series are strictly or too strongly deterministic. For example, if all or even a significant majority of values in a series are ranked, then the theorem does not work for them, or rather decision making for such series will be exactly the opposite of the theorem. The simplest example is the prevalence of an uptrend or a downtrend with some pullbacks.

 

Yuri, why is there still no proof of your "theorem" on the random number generator? Five minutes and all the enemies are defeated. Do you enjoy savoring the end? You're being clever as a scientist, why don't you do a proper experiment as a scientist?

Also very interesting, Yuri, what is the difference between a random series and an arbitrary series as you see it?

 
and depending on whether the numbers are random or non-random, this is a very interesting place would you mind commenting on????
 
Yury Reshetov:

If at least two other random values in a random camp are known. But the point is that determinism is not strict, but probabilistic.

I think it's not hard to give an example of a series that looks random and has no relationships at lag 1, but the value is statistically related to values at other lags whose number >= 1.

But it will be a synthetic series with a known pattern in advance.

If I understand you correctly, I agree that checking for a relationship at one lag is not a sufficient condition for accepting the null hypothesis that the realizations of a random variable are independent of the past. Dependence, in a particular case, can also manifest itself in the fact that a combination of values on lags, for example +1 +2 +3 will be statistically (stochastically) related to a combination on lags - 15 -20 -30.

For example, if the values on three arbitrary lags add up to an even number (and this happens 50% of the time), then the sum of the values on the other three lags will give an even number with a 35% probability. And vice versa. Finding relationships in any pairwise combination of lags will give a p-value within the confidence interval.

 
Alexey Burnakov:


Do I understand correctly that by the theorem, any random series (not explicitly deterministic in any way) will have a dependence on two lags with index i > 1?

Once again, nondeterminism is required such that for any i and j: p(Xi > Xj) = p(Xi < Xj). That is, in a random series (or stream) no single preceding value affects the following one (there is no first level depth consequence)


In such a case, if we add another index, e.g. k (another level), or even several more, the nondeterminism will diminish and the consequence on the depth of the second level becomes evident, since:

p(Xi > Xk | Xi < Xj) ≥ p(Xi < Xj)

Where:

p(A) is the unconditional probability of event A occurring without taking into account additional factors;

p(B | A) is the conditional probability of event A occurring, assuming that event B has already occurred, i.e. taking into account one more factor, event B.

 
Alexey Burnakov:

For example, if the values on three arbitrary lags add up to an even number (and this happens 50% of the time), then the sum of the values on the other three lags will give an even number with 35% probability. And vice versa. In this case, looking for connections in any pairwise combination of lags will give a p-value within the confidence interval.

The theorem is useless here, because even and odd numbers are not pairwise ranked. I.e:

  1. An even number may be greater than or equal to an odd number.
  2. An even number can be greater than another even number, or less than, or equal to it.
  3. An odd number may be greater than, less than, or equal to another odd number.
 
Denis Timoshin:
and depending on whether the numbers are random or not, this is a very interesting place to comment on????

If the value of a quantity cannot be subjectively determined, then that quantity is random.

For example, take playing cards, say a 52 card deck. They all have values from 2 to Ace. If the cards are laid face up we can objectively determine their value. If the cards are face up then the value of any random card is subjectively random to us. However, for a cheat, multiple cards may be subjectively non-random, even though they are face up in relation to the cheat as well.

 
Yury Reshetov:

If the value of a quantity cannot be subjectively determined then it is random.

For example, take playing cards, say a 52 card deck. They all have values from 2 to Ace. If the cards are dealt face up then we can objectively determine their value. If the cards are face up then the value of any random card is subjectively random to us. However, for a cheat, the cards are not subjectively random, even though they are face up in relation to the cheat as well.

Now I understand. Thank you for the full explanation.
 
Denis Timoshin:
now I see. thanks for the full explanation.
It is not a full explanation, because the objectivity of determinism is only a hypothesis.
 
I think it is not correct to classify Forex market as a random process, for the simple reason that it is related to economic processes which have a natural manifestation. We have to look for regularities, which are characteristic to Forex market, but trying to classify it as random, I consider it a defeatist attitude, if not to say even harsher.
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