Theorem on the presence of memory in random sequences - page 29
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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 previous value affects the subsequent 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 apparent, since:
p(Xi > Xk | Xi < Xj) ≥ p(Xi < Xj)
Where:
p(A) is the unconditional probability of the occurrence of event A 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.
I think that 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 regular manifestation. We have to look for regularities, which are characteristic to Forex market, but I consider the attempt to classify it as chance to be a defeatist attitude, if not to say even harsher.
The water flow of a river naturally aims to reach the level of the world's oceans, but the riverbed just as naturally has many random and unexpected turns both to the left and to the right.
It is only natural for a river flow to reach ocean level, but it is also natural for a river bed to have many random and unexpected turns both to the left and to the right.
Water always finds a hole © Folk proverb
The flow of rivers is not random at all, but obeys the laws of least resistance.
Once upon a time I was interested in gold mining. Probably for many it is not a secret that geological prospecting of gold-bearing lodes is most effective through examination of sand in the rivers, i.e. a geologist or prospector moves along the river against the flow. So the most interesting thing is that gold moves along the river bed along the path of the least resistance, i.e. in zigzags due to its high density. Therefore sand for presence of gold grains is taken for a sample before river turns in the course of a stream - there where it can most likely be stuck in the river ground. Looking elsewhere is simply useless.
Moreover, the water flow depends on many influences, which are investigated and not silenced, while in Forex the influences are not less, and a lot of underlying, which cannot be analysed or logically, as well as common sense, is silenced!
Again, water movements are subjectively observable and market movements are hidden under the veil of commercial secrecy. And so the owners of insider information, like cheats in gambling, have a distinct advantage over everyone else.
In the signal, two positions closed in the negative, one reversed to the positive:
In that case if we add another index such as k (another level), or even several more, the nondeterminism will diminish and an aftereffect on the depth of the second level becomes apparent, since:
p(Xi > Xk | Xi < Xj) ≥ p(Xi < Xj)
Where:
p(A) is unconditional probability of event A occurrence without taking into account additional factors;
p(B | A) is the conditional probability of event A occurring, assuming event B has already occurred, i.e. taking into account one more factor, event B.
Yuri, why does this inequality work at all? I can't quite figure it out, I think it's about the characteristics of the series you have in mind.
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)
It's kind of self-explanatory. For example, a series with mu = 0, sd = 1 and no dependencies on adjacent lags would work as an example?
Yuri, why does this inequality even work?
The trailer is a fresh revision of the theorem. At the beginning there is an added problem when a dice is rolled twice, and then the player has to play for rise or fall, i.e. to get the difference between the last result of rolling the dice and the future result. To make it clearer, attached to the problem there is a table with all 216 possible outcomes of triple rolls of the dice, by which you can easily calculate the positive expected value for the player.
Well, and after the problem, there is a detailed analysis with the proof of the very inequality you are asking about?
There are smaller tables - only six lines each. So, it won't be hard to figure it out, if you have the appropriate knowledge of mathematics of course.
The trailer contains a fresh revision of the theorem. In the beginning there is an added problem, when a dice is rolled twice, and then the player has to play for rise or fall, i.e. to get the difference between the last result of rolling and the future result. To make it clearer, attached to the problem there is a table with all 216 possible outcomes of triple rolls of the dice, by which you can easily calculate the positive expected value for the player.
Well, and after the problem, there is a detailed analysis with a proof of the very inequality you are asking about?
There are smaller tables - only six lines each. So, it won't be hard to figure it out, if you have the appropriate knowledge of mathematics of course.