If we have a process whose observed single appearance (n=1) has Poisson statistics:
\( P= \lambda*e^{-\lambda} \)
Then, it can be either result of simple Poisson process, or, process whose logarithm has Poisson distribution (and some mechanism hidden to us performs logarithm operation on input process) - so called log Poisson process: (such processes have recently been involved in models of turbulence (Extended Self Similarity) and Multiplicative ( chain, cascade) stochastic processes).
\( ln(W) = \lambda*e^{-\lambda} \)
The obvious distribution for process W itself is then (without norming , just to see the functional form) :
\( W = e^{{\lambda}*e^{-\lambda}} \)
if we substiture then \( e^{\lambda} = z \) the log-Poisson process has distribution:
\( W=z^{(1/z)} \) -self root which is so familiar from infinite tetration as it is its inverse!
\( h(W) = h(z^{(1/z)}) = z \)
So we can say that:
Self -root represents a distribution for a log-Poisson process W itself (unnormed-I do not know how to do norming) ( process W whose logarithm has Poisson distribution);
What is infinite tetration then? Since infinite tetration work on both z^(1/z) and (1/z)^z returning either z or 1/z , we can say that infinite tetration returns a pair of conjugate values linked to log-Poisson process , so that log-Poisson process underlying \( W \) distribution is split into 2 parts:
\( z= e^{\lambda} \) and \( 1/z= e^{-\lambda} \)
From these we can see that these conjugate values lead to a pair of growth/decay process..They are also "memoryless" processes, each of them in opposite direction ( if we consider \( \lambda \) as "time" - memoryless towards past and predictionless towards future ( a well known property of exponential process- whereever You are on it, it looks the same).
I guess that if we had a process which randomly grows as e^x and has a random instantenous feedback e^-x in each scale (value) x (and vice versa) than it could end up with log-Poisson distribution or distribution characterized by a pair of conjugate values.
The above can be extended to \( \lambda \) being imaginary and complex, thus leading to a notion that underlying math behind complex mathematics is actually stochastic, random, and e.g. \( I=e^{I*pi/2} ,-I= e^{-I*pi/2 } \) are exactly such a pair of growth/decay speed values, more so, the distribution \( W \) in this case is invariant relative to order of growth/decay process- any order as long as they oscillate between themselves will lead to the same distribution- which is characterized in real domain by its value, \( I^{(1/I)}=(1/I)^I= e^{\pi/2}=4,81... \).
Which in turn may explain the true reason behind amazing correlation between 2D turbulence and conformality of functions of complex argument.
Ivars
\( P= \lambda*e^{-\lambda} \)
Then, it can be either result of simple Poisson process, or, process whose logarithm has Poisson distribution (and some mechanism hidden to us performs logarithm operation on input process) - so called log Poisson process: (such processes have recently been involved in models of turbulence (Extended Self Similarity) and Multiplicative ( chain, cascade) stochastic processes).
\( ln(W) = \lambda*e^{-\lambda} \)
The obvious distribution for process W itself is then (without norming , just to see the functional form) :
\( W = e^{{\lambda}*e^{-\lambda}} \)
if we substiture then \( e^{\lambda} = z \) the log-Poisson process has distribution:
\( W=z^{(1/z)} \) -self root which is so familiar from infinite tetration as it is its inverse!
\( h(W) = h(z^{(1/z)}) = z \)
So we can say that:
Self -root represents a distribution for a log-Poisson process W itself (unnormed-I do not know how to do norming) ( process W whose logarithm has Poisson distribution);
What is infinite tetration then? Since infinite tetration work on both z^(1/z) and (1/z)^z returning either z or 1/z , we can say that infinite tetration returns a pair of conjugate values linked to log-Poisson process , so that log-Poisson process underlying \( W \) distribution is split into 2 parts:
\( z= e^{\lambda} \) and \( 1/z= e^{-\lambda} \)
From these we can see that these conjugate values lead to a pair of growth/decay process..They are also "memoryless" processes, each of them in opposite direction ( if we consider \( \lambda \) as "time" - memoryless towards past and predictionless towards future ( a well known property of exponential process- whereever You are on it, it looks the same).
I guess that if we had a process which randomly grows as e^x and has a random instantenous feedback e^-x in each scale (value) x (and vice versa) than it could end up with log-Poisson distribution or distribution characterized by a pair of conjugate values.
The above can be extended to \( \lambda \) being imaginary and complex, thus leading to a notion that underlying math behind complex mathematics is actually stochastic, random, and e.g. \( I=e^{I*pi/2} ,-I= e^{-I*pi/2 } \) are exactly such a pair of growth/decay speed values, more so, the distribution \( W \) in this case is invariant relative to order of growth/decay process- any order as long as they oscillate between themselves will lead to the same distribution- which is characterized in real domain by its value, \( I^{(1/I)}=(1/I)^I= e^{\pi/2}=4,81... \).
Which in turn may explain the true reason behind amazing correlation between 2D turbulence and conformality of functions of complex argument.
Ivars