I introduce here a family of very peculiar statistical distributions governed by two parameters: p<\/em>, a real number in [0, 1], and b<\/em>, an integer > 1. These distributions were discovered by solving the following functional equation, corresponding to b<\/em> = 2.\u00a0<\/p>\n Here\u00a0f<\/em>(x<\/em>) is the density attached to that distribution. The support domain for x<\/em> is also [0, 1]. This type of distribution appears in the following context.\u00a0<\/p>\n Let Z<\/i>\u00a0be an irrational number in [0, 1] (called\u00a0seed<\/em>) and consider the sequence x<\/em>(n<\/em>) = {b<\/em>^n<\/em>\u00a0Z<\/i>}. Here the brackets represent the fractional part function. In particular,\u00a0INT(b<\/em>\u00a0x<\/em>(n<\/em>)) is the\u00a0n<\/span><\/span><\/span><\/em><\/span>-th digit of Z<\/i>\u00a0<\/span>in base b<\/em>.\u00a0<\/span>The values\u00a0x<\/em>(n<\/em>)<\/span>\u00a0are distributed in a certain way due to the\u00a0ergodicity<\/em>\u00a0of the underlying process. The density associated with this distribution is the function\u00a0f<\/span><\/em><\/span>, and for the immense majority of seeds Z<\/i>,\u00a0<\/span>that density is uniform on [0, 1].\u00a0Seeds\u00a0<\/span>producing the uniform density are sometimes called\u00a0normal<\/em>\u00a0numbers; their digit distribution is also uniform.<\/p>\n However, the functional equation 2f<\/em>(x<\/em>) =\u00a0f<\/em>(x<\/em>\/2) + f<\/em>((1+x<\/em>)\/2) may have plenty of other solutions. Such solutions are called non-standard<\/em> solutions. The set of seeds producing non-standard solutions is known to have Lebesgue measure zero, but there are infinitely many such seeds.\u00a0All rational seeds are, but they produce a discrete distribution. Thus their density is of the discrete type. We are interested here in a non-discrete solution.\u00a0<\/p>\n 1. Example with p<\/em> = 0.75 and b<\/em> = 2<\/strong><\/p>\n The uniform distribution corresponds to p<\/em> = 0.5. Below is a non-standard density satisfying the requirements. Actually, the plot below represents its percentile distribution. It was produced with a seed Z<\/i>\u00a0in [0,1] built as follows: the\u00a0n<\/em>-th binary digit of Z<\/em> is 1 if Rand(n<\/em>)\u00a0 <\u00a0\u00a0p<\/em>, and 0 otherwise, using a pseudo random number generator. Here p<\/em> = 0.75. Note that P.25 = 0.5 and corresponds to a dip in the chart below (P.25 denotes the 25-th<\/em> percentile.) Dips are everywhere, only the big ones are visible. By contrast, the percentile distribution for the uniform (standard) case p<\/em> = 0.5\u00a0is a straight line, with no dips.<\/p>\n 2. General solution<\/strong><\/p>\n The functional equation is a bit more complicated if b<\/em> is not equal to 2. It becomes<\/p>\n Using the construction mechanism outlined in the previous section to generate a non-standard seed Z<\/i>\u00a0(sometimes called a non-normal number<\/a> or bad seed<\/em>), it is clear that x<\/em>(n<\/em>) is a random variable. We also have Take the derivative of the inverse Fourier transform (see section\u00a0<\/span>inverse formula<\/em>\u00a0<\/span>here<\/a>) and you obtain<\/span><\/p>\n If p<\/em> = 0.5 and b<\/em> = 2 we are back to the uniform case. Otherwise the solution is quite special: the density f<\/em> is nowhere differentiable it seems. See picture below for p<\/em> = 0.55 and b<\/em> = 2.<\/span><\/p>\n Now we should prove that this case is ergodic<\/em>, for the functional equation to apply. I also tried to check with some sampled values of x<\/em> to see whether\u00a02f<\/em>(x<\/em>) =\u00a0f<\/em>(x<\/em>\/2) + f<\/em>((1+x<\/em>)\/2)<\/span>, but the function being discontinuous everywhere, and since I got its value approximated probably to no more than two decimals, it is not easy.<\/span><\/p>\n 3. Applications, properties and data<\/strong><\/p>\n The distribution attached to this type of density has the following moments:<\/span><\/p>\n Why does f<\/em>(x<\/em>) must satisfy the functional equation discussed above? This a consequence of the fact that the underlying distribution is the equilibrium distribution for the sequence x<\/em>(n<\/em>) = {b<\/em> x<\/em>(n<\/em>-1) } = {b<\/em>^n<\/em> Z<\/em>}. In particular, the equilibrium distribution is solution to some stochastic integral equation P(X<\/i>\u00a0< x) = P({b<\/em> X<\/em>}\u00a0 <\u00a0\u00a0x<\/em>).\u00a0 For details, see my book\u00a0Applied Stochastic Processes, Chaos Modeling, and Probabilistic Properties of Numeration Systems<\/em>\u00a0available here<\/a>, see pages 65-66.<\/p>\n Potential applications are found in cryptography, Fintech (stock market modeling), Bitcoin, number theory, random number generation, benchmarking statistical tests (see here<\/a>) and even gaming (see here<\/a>.) However, the most interesting application is probably to gain insights about how non-normal numbers look like, especially their chaotic nature. It is a fundamental tool to help solve one of the most intriguing mathematical conjectures of all times (yet unsolved): are the digits of standard constants such as Pi or SQRT(2) uniformly distributed or not? For instance, when b<\/em> = 2, any departure from p<\/em> = 0.5 (a normal seed) results in a strong discontinuity for f<\/em>(x<\/em>) at x<\/em> = 0.5. If you look at the above chart, f(<\/em>0) = f(<\/em>1\/2) = f<\/em>(1) regardless of p<\/em>, but discontinuities are masking this fact.\u00a0<\/p>\n The charts featured here, as well as the underlying computations, were all produced in Excel. You can download the spreadsheet here<\/a>. In particular, a very efficient algorithm is used to produce (say) one million digits of Z, and to compute one million successive values of x<\/em>(n<\/em>) each with a precision of 14 decimals. You can play interactively with the parameters b<\/em> and p<\/em> in the spreadsheet, and even try non-integer values of b<\/em> (I suggest you try b<\/em> = 1.5 and p<\/em> = 0.5). If b<\/em>\u00a0 < 2 is not an integer, the functional equation is more complicated: it is found in section 2.1\u00a0in this article<\/a>.\u00a0<\/p>\n<\/p>\n<\/div>\n Go to Source<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<\/a><\/p>\n
<\/a><\/p>\n
<\/a><\/p>\n
<\/a>where b<\/em> is the base and d<\/em>(n<\/em>+k<\/em>) is the (n<\/em>+k<\/em>)-th digit of the seed Z<\/i>\u00a0in base b<\/em>.\u00a0 This formula is very useful for computations. Note that Z<\/i>\u00a0= x<\/em>(0).\u00a0Furthermore, by construction, these digits are identically and independently distributed with a Bernouilli distribution of parameter p<\/em>. Thus, using the convolution theorem<\/a>, the characteristic function<\/a> for the seed Z<\/i>\u00a0is<\/p>\n
<\/a><\/p>\n
<\/a><\/span><\/p>\n
<\/a><\/span><\/p>\n
\n