# Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a Numerically Verifiable Infinite Variance

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### Organization of the Work

## 2. Benci’s Alpha-Theory and the Euclidean Numbers

**Axiom 1. Axiom of Archimedes.**

- AT is an introduction to Non-Standard Analysis based on the notion of $\alpha $-limit. The notion of $\alpha $-limit is a version of the transfer principle easier to be used by practitioners; in fact, roughly speaking it could be enunciated as follows: “every relation between sequences is preserved by the limit”
- The theory of numerosity is strictly related to AT. It is useful to give a meaning to some infinite numbers such as the number $\alpha $ which is the numerosity of the set of positive natural numbers and it can be useful in some applications, e.g., see [15].

#### Benci’s Alpha-Theory

**Axiom**

**2.**

**Definition**

**1.**

- ξ is infinite ⟺$\forall \phantom{\rule{0.166667em}{0ex}}n\in \mathbb{N},\left|\xi \right|>n$
- ξ is finite ⟺$\exists \phantom{\rule{0.166667em}{0ex}}n\in \mathbb{N}$, $\frac{1}{n}}<\left|\xi \right|<n$
- ξ is infinitesimal $\u27fa\forall \phantom{\rule{0.166667em}{0ex}}n\in \mathbb{N},$$\left|\xi \right|<{\textstyle \frac{1}{n}}\phantom{\rule{0.166667em}{0ex}}.$

**Axiom**

**3.**

- if A is finite $\mathfrak{num}\left(A\right)=\left|A\right|\phantom{\rule{0.277778em}{0ex}}$($|\xb7|$ denotes the cardinality of a set)
- $\mathfrak{num}\left(A\right)<\mathfrak{num}\left(B\right)$ if $A\subset B$
- $\mathfrak{num}(A\cup B)=\mathfrak{num}\left(A\right)+\mathfrak{num}\left(B\right)-\mathfrak{num}(A\cap B)$
- $\mathfrak{num}(A\times B)=\mathfrak{num}\left(A\right)\xb7\mathfrak{num}\left(B\right)$
- $\alpha =\mathfrak{num}\left(\mathbb{N}\right)\phantom{\rule{0.166667em}{0ex}}.$

**Axiom**

**4.**

- 1.
- if $\xi \in \mathbb{E}$, then there exists a sequence $\phi :\mathbb{N}\to \mathbb{R}$ such that$$\xi =\underset{n\uparrow \alpha}{lim}\phi \left(n\right)$$
- 2.
- if $\phi \left(n\right)=n,$ then$$\underset{n\uparrow \alpha}{lim}\phi \left(n\right)=\alpha $$
- 3.
- if eventually $\phi \left(n\right)\ge \psi \left(n\right)$ (namely $\exists \phantom{\rule{0.166667em}{0ex}}{n}_{0}\in \mathbb{N}$ such that $\forall \phantom{\rule{0.166667em}{0ex}}n\ge {n}_{0},\phantom{\rule{4pt}{0ex}}\phi \left(n\right)\ge \psi \left(n\right)$), then$$\underset{n\uparrow \alpha}{lim}\phi \left(n\right)\ge \underset{n\uparrow \alpha}{lim}\psi \left(n\right)$$
- 4.
- for every sequence $\phantom{\rule{4pt}{0ex}}\phi ,\psi $$$\begin{array}{cc}\hfill \underset{n\uparrow \alpha}{lim}\phi \left(n\right)+\underset{n\uparrow \alpha}{lim}\psi \left(n\right)& =\underset{n\uparrow \alpha}{lim}\left(\phi \left(n\right)+\psi \left(n\right)\right)\hfill \\ \hfill \underset{n\uparrow \alpha}{lim}\phi \left(n\right)\xb7\underset{n\uparrow \alpha}{lim}\psi \left(n\right)& =\underset{n\uparrow \alpha}{lim}\left(\phi \left(n\right)\xb7\psi \left(n\right)\right)\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$

**Theorem**

**1.**

**Standard part**There exists a function

- $x\sim y\Rightarrow st\left(x\right)=st\left(y\right)$
- $st\left(st\right(x\left)\right)=st\left(x\right)$

**Theorem**

**2.**

- Divisibility Property: For every $k\in \mathbb{N}$, the number α is a multiple of k and the numerosity of the set of multiples of k:$$\mathfrak{num}\left(\{k,2k,3k,\dots ,nk,\dots \}\right)=\frac{\alpha}{k}\phantom{\rule{0.166667em}{0ex}}.$$
- Root Property: For every $k\in \mathbb{N}$, the number α is a k-th power and the numerosity of the set of k-th powers:$$\mathfrak{num}\left(\{{1}^{k},{2}^{k},{3}^{k},\dots ,{n}^{k},\dots \}\right)=\sqrt[k]{\alpha}\phantom{\rule{0.166667em}{0ex}}.$$
- Power Property: If we set ${\mathcal{P}}_{fin}\left(A\right)=\{F\in \mathcal{P}\left(A\right)\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}F\phantom{\rule{4pt}{0ex}}$ is a finite set}, then$$\mathfrak{num}\left({\mathcal{P}}_{fin}\left({\mathbb{N}}^{+}\right)\right)={2}^{\alpha}\phantom{\rule{0.166667em}{0ex}}.$$
- Integer numbers Property:$$\mathfrak{num}\left(\mathbb{Z}\right)=2\alpha +1\phantom{\rule{0.166667em}{0ex}}.$$
- Rational numbers Property: For every $q\in \mathbb{Q}$$$\mathfrak{num}\left(\mathbb{Q}\right)=2{\alpha}^{2}+1\phantom{\rule{0.166667em}{0ex}},$$and$$\mathfrak{num}(\left(q,q+1\right]\cap \mathbb{Q})=\mathfrak{num}(\left(0,1\right]\cap \mathbb{Q})=\alpha \phantom{\rule{0.166667em}{0ex}}.$$

**Proof.**

## 3. The Algorithmic Numbers

**Definition**

**2.**

**Algorithmic number**A number $\xi \in \mathbb{E}$ is called algorithmic if it can be represented as a finite sum of monosemia, namely

- the inverse of an AN is not always an AN, e.g., ${\left(\alpha +1\right)}^{-1}$ is not an AN;
- they have a variable length coding, and hence they are not suitable to number crunching-oriented applications.

#### 3.1. The BANs (Bounded Algorithmic Numbers)

**Sum**(assuming $p\ge q$):

**Product**:

**Division**:

#### 3.2. A Summary: Real Numbers vs. Euclidean Numbers

## 4. A New Definition of Heavy Tailed Distribution

**New Definition of heavy tailed distributions:**A distribution is considered heavy tailed if its variance diverges towards $+\infty $, or if it assumes a (concrete) infinite value.

## 5. The Euclidean Gaussian Distribution

#### 5.1. How to Generate Pseudo-Random Numbers According to It

`Ban`and

`BanArray`, to handle scalar and vector quantities, respectively.

`BanArray`, constructed as a column vector of N (where N is obviously the required size of the

`BanArray`) rows of

`Ban`, according to the desired Gaussian probability density. To do this, we start from the realisations of a standard Gaussian distribution with zero mean and unit variance, obtained simply with the built-in function

`randn(N,1)`, and we go to multiply each realisation, one by one, by $\sigma $ (i.e., the theoretical standard deviation, which this time is a

`Ban`, i.e., a Euclidean number), to obtain the desired variance, and to sum element-wise for the set mean (which can again be a Euclidean number).

`Ban`used, i.e., the degree of the polynomial in $\eta $ of the

`Ban`, and the theoretical values of mean and standard deviation that we want to use for the Gaussian probability density to be generated. The leading exponent is set at will through an input parameter in the call to the constructor of the

`BanArray`class.

#### 5.2. How to Numerically Verify the Sample Mean and Variance

`est_mean`) and the estimated variance (

`est_var`) is entrusted, respectively, to the two Matlab functions

`mean`and

`var`, that we developed in order to exploit the classical sample definitions of mathematical statistics, appropriately overloaded for the

`BanArray`class:

`est_mean = mean(x); % x is a BanArray, est_mean is a Ban`

`est_var = var(x); % x is a BanArray, est_var is a Ban`

## 6. Euclidean LogNormal Distributions

#### 6.1. A Euclidean LogNormal with Finite Mean and Infinite Variance

#### 6.2. How to Numerically Assess whether the Euclidean LogNormal Distribution Has the Desired Mean and Variance

#### 6.3. An Observation about Geometric and Harmonic Means

- an infinitesimal theoretical geometric mean: $\stackrel{\u02d8}{GM}\left\{X\right\}={e}^{-\frac{{\alpha}^{2}}{2}}\phantom{\rule{0.166667em}{0ex}},$
- a finite theoretical mean: $\stackrel{\u02d8}{E}\left\{X\right\}=1\phantom{\rule{0.166667em}{0ex}},$
- an infinite theoretical variance: $\stackrel{\u02d8}{VAR}\left\{X\right\}={e}^{{\alpha}^{2}}-1\phantom{\rule{0.166667em}{0ex}}.$

#### 6.4. Generalizing our Euclidean LogNormal: A 3-Parameter Version

## 7. Conclusions

`Ban`and

`BanArray`classes. Then we have used them to design a LogNormal distribution having a finite mean and an infinite variance, and to numerically verify that the pseudo-random numbers generated according to this function are exactly (apart from numerical estimation errors) the ones predicted by the theory, and the numerical estimation error decreases as the number of samples increases.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

VAR | Variance |

BAN | Bounded Algorithmic Number |

Ban | Bounded Algorithmic Number (name of the Matlab class) |

BanArray | Vector of Bounded Algorithmic Numbers (name of the Matlab class) |

GM | Geometric Mean |

HM | Harmonic Mean |

AM | Arithmetic Mean |

## Appendix A. An Example of the Samples Generated for a Euclidean Gaussian

**Table A1.**Pseudo-random samples related to the Euclidean Gaussian considered in Section 5.2, and associated to Table 2, for the case ${10}^{2}$ samples. We recall that its theoretical mean is $(-1.5+20.33\eta -403.26{\eta}^{2}){\alpha}^{1}$, its theoretical variance is $(11.56-97.444\eta -768.853{\eta}^{2})$, while the estimated mean is $(-1.5+19.8637\eta -400.701{\eta}^{2}){\alpha}^{1}$ and its estimated variance is $(13.7816-114.018\eta +246.011{\eta}^{2})$ (both estimated using the 100 samples below).

Sample # | X | Sample # | X |
---|---|---|---|

1 | $(-1.5+22.4962\eta -416.787{\eta}^{2}){\alpha}^{1}$ | 51 | $(-1.5+23.8709\eta -417.318{\eta}^{2}){\alpha}^{1}$ |

2 | $(-1.5+25.2201\eta -426.399{\eta}^{2}){\alpha}^{1}$ | 52 | $(-1.5+25.0048\eta -420.661{\eta}^{2}){\alpha}^{1}$ |

3 | $(-1.5+21.6425\eta -413.680{\eta}^{2}){\alpha}^{1}$ | 53 | $(-1.5+20.6005\eta -404.389{\eta}^{2}){\alpha}^{1}$ |

4 | $(-1.5+21.8335\eta -409.761{\eta}^{2}){\alpha}^{1}$ | 54 | $(-1.5+14.7786\eta -385.606{\eta}^{2}){\alpha}^{1}$ |

5 | $(-1.5+18.6227\eta -396.666{\eta}^{2}){\alpha}^{1}$ | 55 | $(-1.5+22.9006\eta -413.423{\eta}^{2}){\alpha}^{1}$ |

6 | $(-1.5+19.1047\eta -399.796{\eta}^{2}){\alpha}^{1}$ | 56 | $(-1.5+18.9222\eta -393.984{\eta}^{2}){\alpha}^{1}$ |

7 | $(-1.5+16.3357\eta -385.383{\eta}^{2}){\alpha}^{1}$ | 57 | $(-1.5+22.9960\eta -411.908{\eta}^{2}){\alpha}^{1}$ |

8 | $(-1.5+15.6063\eta -380.955{\eta}^{2}){\alpha}^{1}$ | 58 | $(-1.5+8.97068\eta -358.333{\eta}^{2}){\alpha}^{1}$ |

9 | $(-1.5+20.3847\eta -405.783{\eta}^{2}){\alpha}^{1}$ | 59 | $(-1.5+22.5302\eta -407.495{\eta}^{2}){\alpha}^{1}$ |

10 | $(-1.5+22.1259\eta -405.929{\eta}^{2}){\alpha}^{1}$ | 60 | $(-1.5+19.0771\eta -392.468{\eta}^{2}){\alpha}^{1}$ |

11 | $(-1.5+20.8233\eta -409.860{\eta}^{2}){\alpha}^{1}$ | 61 | $(-1.5+18.4388\eta -394.555{\eta}^{2}){\alpha}^{1}$ |

12 | $(-1.5+14.2839\eta -376.800{\eta}^{2}){\alpha}^{1}$ | 62 | $(-1.5+18.3367\eta -386.855{\eta}^{2}){\alpha}^{1}$ |

13 | $(-1.5+24.5148\eta -415.332{\eta}^{2}){\alpha}^{1}$ | 63 | $(-1.5+21.2766\eta -402.290{\eta}^{2}){\alpha}^{1}$ |

14 | $(-1.5+16.8114\eta -389.671{\eta}^{2}){\alpha}^{1}$ | 64 | $(-1.5+16.8916\eta -392.727{\eta}^{2}){\alpha}^{1}$ |

15 | $(-1.5+20.7545\eta -406.961{\eta}^{2}){\alpha}^{1}$ | 65 | $(-1.5+21.6471\eta -410.025{\eta}^{2}){\alpha}^{1}$ |

16 | $(-1.5+19.0400\eta -399.484{\eta}^{2}){\alpha}^{1}$ | 66 | $(-1.5+13.9704\eta -375.993{\eta}^{2}){\alpha}^{1}$ |

17 | $(-1.5+16.1227\eta -386.673{\eta}^{2}){\alpha}^{1}$ | 67 | $(-1.5+18.6799\eta -394.883{\eta}^{2}){\alpha}^{1}$ |

18 | $(-1.5+28.1959\eta -437.504{\eta}^{2}){\alpha}^{1}$ | 68 | $(-1.5+18.7584\eta -392.743{\eta}^{2}){\alpha}^{1}$ |

19 | $(-1.5+18.0416\eta -392.397{\eta}^{2}){\alpha}^{1}$ | 69 | $(-1.5+12.5249\eta -370.446{\eta}^{2}){\alpha}^{1}$ |

20 | $(-1.5+22.1598\eta -414.842{\eta}^{2}){\alpha}^{1}$ | 70 | $(-1.5+22.8677\eta -417.810{\eta}^{2}){\alpha}^{1}$ |

21 | $(-1.5+24.7817\eta -419.190{\eta}^{2}){\alpha}^{1}$ | 71 | $(-1.5+26.9057\eta -426.264{\eta}^{2}){\alpha}^{1}$ |

22 | $(-1.5+15.8078\eta -385.981{\eta}^{2}){\alpha}^{1}$ | 72 | $(-1.5+19.7376\eta -396.241{\eta}^{2}){\alpha}^{1}$ |

23 | $(-1.5+20.4450\eta -401.143{\eta}^{2}){\alpha}^{1}$ | 73 | $(-1.5+18.5704\eta -395.276{\eta}^{2}){\alpha}^{1}$ |

24 | $(-1.5+21.2017\eta -402.164{\eta}^{2}){\alpha}^{1}$ | 74 | $(-1.5+12.6691\eta -376.308{\eta}^{2}){\alpha}^{1}$ |

25 | $(-1.5+20.4024\eta -408.407{\eta}^{2}){\alpha}^{1}$ | 75 | $(-1.5+16.8693\eta -389.817{\eta}^{2}){\alpha}^{1}$ |

26 | $(-1.5+19.6540\eta -398.629{\eta}^{2}){\alpha}^{1}$ | 76 | $(-1.5+19.0667\eta -394.974{\eta}^{2}){\alpha}^{1}$ |

27 | $(-1.5+20.2672\eta -407.769{\eta}^{2}){\alpha}^{1}$ | 77 | $(-1.5+23.6460\eta -414.981{\eta}^{2}){\alpha}^{1}$ |

28 | $(-1.5+18.0191\eta -391.469{\eta}^{2}){\alpha}^{1}$ | 78 | $(-1.5+15.9128\eta -381.605{\eta}^{2}){\alpha}^{1}$ |

29 | $(-1.5+20.3515\eta -400.110{\eta}^{2}){\alpha}^{1}$ | 79 | $(-1.5+22.5581\eta -415.944{\eta}^{2}){\alpha}^{1}$ |

30 | $(-1.5+22.1714\eta -407.758{\eta}^{2}){\alpha}^{1}$ | 80 | $(-1.5+22.7076\eta -416.133{\eta}^{2}){\alpha}^{1}$ |

31 | $(-1.5+17.9911\eta -391.271{\eta}^{2}){\alpha}^{1}$ | 81 | $(-1.5+13.0442\eta -367.553{\eta}^{2}){\alpha}^{1}$ |

32 | $(-1.5+20.0107\eta -399.708{\eta}^{2}){\alpha}^{1}$ | 82 | $(-1.5+18.5946\eta -396.663{\eta}^{2}){\alpha}^{1}$ |

33 | $(-1.5+24.2437\eta -421.933{\eta}^{2}){\alpha}^{1}$ | 83 | $(-1.5+18.2342\eta -392.373{\eta}^{2}){\alpha}^{1}$ |

34 | $(-1.5+15.2698\eta -384.007{\eta}^{2}){\alpha}^{1}$ | 84 | $(-1.5+18.8147\eta -394.900{\eta}^{2}){\alpha}^{1}$ |

35 | $(-1.5+17.5150\eta -388.349{\eta}^{2}){\alpha}^{1}$ | 85 | $(-1.5+18.8560\eta -395.944{\eta}^{2}){\alpha}^{1}$ |

36 | $(-1.5+21.9145\eta -406.013{\eta}^{2}){\alpha}^{1}$ | 86 | $(-1.5+17.7508\eta -393.736{\eta}^{2}){\alpha}^{1}$ |

37 | $(-1.5+20.9541\eta -403.053{\eta}^{2}){\alpha}^{1}$ | 87 | $(-1.5+20.4584\eta -402.923{\eta}^{2}){\alpha}^{1}$ |

38 | $(-1.5+19.0133\eta -399.941{\eta}^{2}){\alpha}^{1}$ | 88 | $(-1.5+20.5856\eta -398.918{\eta}^{2}){\alpha}^{1}$ |

39 | $(-1.5+15.4042\eta -390.100{\eta}^{2}){\alpha}^{1}$ | 89 | $(-1.5+28.1696\eta -436.693{\eta}^{2}){\alpha}^{1}$ |

40 | $(-1.5+15.8819\eta -383.715{\eta}^{2}){\alpha}^{1}$ | 90 | $(-1.5+12.2121\eta -367.368{\eta}^{2}){\alpha}^{1}$ |

41 | $(-1.5+22.4115\eta -406.794{\eta}^{2}){\alpha}^{1}$ | 91 | $(-1.5+25.8154\eta -427.502{\eta}^{2}){\alpha}^{1}$ |

42 | $(-1.5+22.6872\eta -407.943{\eta}^{2}){\alpha}^{1}$ | 92 | $(-1.5+20.3636\eta -403.066{\eta}^{2}){\alpha}^{1}$ |

43 | $(-1.5+26.5761\eta -425.247{\eta}^{2}){\alpha}^{1}$ | 93 | $(-1.5+19.2718\eta -393.848{\eta}^{2}){\alpha}^{1}$ |

44 | $(-1.5+19.9352\eta -403.147{\eta}^{2}){\alpha}^{1}$ | 94 | $(-1.5+13.0922\eta -366.607{\eta}^{2}){\alpha}^{1}$ |

45 | $(-1.5+18.4954\eta -396.429{\eta}^{2}){\alpha}^{1}$ | 95 | $(-1.5+18.8844\eta -396.287{\eta}^{2}){\alpha}^{1}$ |

46 | $(-1.5+25.9097\eta -429.933{\eta}^{2}){\alpha}^{1}$ | 96 | $(-1.5+23.7794\eta -413.745{\eta}^{2}){\alpha}^{1}$ |

47 | $(-1.5+23.4765\eta -413.668{\eta}^{2}){\alpha}^{1}$ | 97 | $(-1.5+20.4874\eta -405.223{\eta}^{2}){\alpha}^{1}$ |

48 | $(-1.5+16.7999\eta -391.804{\eta}^{2}){\alpha}^{1}$ | 98 | $(-1.5+22.9617\eta -414.269{\eta}^{2}){\alpha}^{1}$ |

49 | $(-1.5+27.1523\eta -431.919{\eta}^{2}){\alpha}^{1}$ | 99 | $(-1.5+18.9252\eta -396.288{\eta}^{2}){\alpha}^{1}$ |

50 | $(-1.5+13.9762\eta -373.639{\eta}^{2}){\alpha}^{1}$ | 100 | $(-1.5+21.5426\eta -412.867{\eta}^{2}){\alpha}^{1}$ |

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Real Numbers | Euclidean Numbers |
---|---|

Can be represented on a computer using a fixed-length representation, such as the 64-bit IEEE 754/2019 standard. | Can be represented on a computer using a fixed-length representation, such as BAN2 (a polynomial in $\eta $ having degree two). |

An example: $\mathbf{2.714}\xb7{\mathbf{10}}^{-\mathbf{8}}$ | An example: $\mathbf{(}\mathbf{5}-\mathbf{7}\mathbf{\eta}\mathbf{+}\mathbf{11}{\mathbf{\eta}}^{\mathbf{2}}\mathbf{)}\xb7{\mathbf{\alpha}}^{\mathbf{-}\mathbf{3}}$ |

Important remark: | Important remark: |

The result of the multiplication of two 64-bit floating point numbers is still a 64-bit floating point number (this is really helpful in numerical computations when using iterative methods, and holds true for the other arithmetic operations as well) | The result of the multiplication of two BAN2 numbers is still a BAN2 number (this is really helpful in numerical computations when using iterative methods, and holds true for the other arithmetic operations as well) |

Convenient ASCII display format: | Convenient ASCII display format: |

2.714E-8 | (5 - 7 11)A-3 |

(where E-8 means ${10}^{-8}$) | (where A-3 means ${\alpha}^{-3}$) |

**Table 2.**Numerical verification for the Euclidean Gaussian having infinite theoretical mean $\mu =(-1.5+20.33\eta -403.26{\eta}^{2}){\alpha}^{1}$ and finite theoretical variance ${\sigma}^{2}=(11.56-97.444\eta +217.044{\eta}^{2})$.

# of Samples | $\mathit{E}\left\{\mathit{X}\right\}$ (Sample Mean) | $\mathit{VAR}\left\{\mathit{X}\right\}$ (Sample Variance) |
---|---|---|

${10}^{2}$ | $(-1.5+19.8637\eta -400.701{\eta}^{2}){\alpha}^{1}$ | $(13.7816-114.018\eta +246.011{\eta}^{2})$ |

${10}^{3}$ | $(-1.5+20.3742\eta -403.674{\eta}^{2}){\alpha}^{1}$ | $(12.0228-101.538\eta +226.115{\eta}^{2})$ |

${10}^{4}$ | $(-1.5+20.3353\eta -403.262{\eta}^{2}){\alpha}^{1}$ | $(11.5572-97.4869\eta +217.224{\eta}^{2})$ |

${10}^{5}$ | $(-1.5+20.3318\eta -403.270{\eta}^{2}){\alpha}^{1}$ | $(11.5690-97.5113\eta +217.045{\eta}^{2})$ |

${10}^{6}$ | $(-1.5+20.3316\eta -403.276{\eta}^{2}){\alpha}^{1}$ | $(11.5695-97.5118\eta +217.043{\eta}^{2})$ |

**Table 3.**Sample Mean and Sample Variance of the Euclidean LogNormal having a finite mean and infinite variance, obtained by setting $\sigma $ = $\alpha $ and $\mu $ = $(-\frac{1}{2}{\alpha}^{2}+9.54681).$ The finite theoretical mean is $\stackrel{\u02d8}{E}\left\{X\right\}$ = ${e}^{9.54681}$, while the infinite theoretical variance is $\stackrel{\u02d8}{VAR}\left\{X\right\}$ = ${e}^{(1+19.0936{\eta}^{2}){\alpha}^{2}}-{e}^{19.0936}$. If the coefficients highlighted in boldface were zero, we would have obtained an almost perfect match between theory and practice.

# | $\mathit{E}\left\{\mathit{X}\right\}$ (Sample Mean) | $\mathit{VAR}\left\{\mathit{X}\right\}$ (Sample Variance) |
---|---|---|

${10}^{3}$ | ${e}^{(\mathbf{0}.\mathbf{0260943}+\mathbf{0}.\mathbf{000490988}\eta +9.55455{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(1.10438+\mathbf{0}.\mathbf{000981975}\eta +19.1091{\eta}^{2}){\alpha}^{2}}-{e}^{(\mathbf{0}.\mathbf{0521886}+\mathbf{0}.\mathbf{000981975}\eta +19.1091{\eta}^{2}){\alpha}^{2}}$ |

${10}^{4}$ | ${e}^{(-\mathbf{0}.\mathbf{0126293}+\mathbf{0}.\mathbf{0190467}\eta +9.56013{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(0.949483+\mathbf{0}.\mathbf{0380934}\eta +19.1203{\eta}^{2}){\alpha}^{2}}-{e}^{(-\mathbf{0}.\mathbf{0252587}+\mathbf{0}.\mathbf{0380934}\eta +19.1203{\eta}^{2}){\alpha}^{2}}$ |

${10}^{5}$ | ${e}^{(\mathbf{0}.\mathbf{00157857}-\mathbf{0}.\mathbf{00351095}\eta +9.54501{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(1.00631-\mathbf{0}.\mathbf{0070219}\eta +19.09{\eta}^{2}){\alpha}^{2}}-{e}^{(\mathbf{0}.\mathbf{00315714}-\mathbf{0}.\mathbf{0070219}\eta +19.09{\eta}^{2}){\alpha}^{2}}$ |

${10}^{6}$ | ${e}^{(-\mathbf{0}.\mathbf{00103881}+\mathbf{0}.\mathbf{000230083}\eta +9.54617{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(0.995845+\mathbf{0}.\mathbf{000460166}\eta +19.0923{\eta}^{2}){\alpha}^{2}}-{e}^{(-\mathbf{0}.\mathbf{00207762}+\mathbf{0}.\mathbf{000460166}\eta +19.0923{\eta}^{2}){\alpha}^{2}}$ |

${10}^{7}$ | ${e}^{(\mathbf{0}.\mathbf{000162514}+\mathbf{0}.\mathbf{000439597}\eta +9.54666{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(1.00065+\mathbf{0}.\mathbf{000879195}\eta +19.0933{\eta}^{2}){\alpha}^{2}}-{e}^{(\mathbf{0}.\mathbf{000325027}+\mathbf{0}.\mathbf{000879195}\eta +19.0933{\eta}^{2}){\alpha}^{2}}$ |

**Table 4.**This table is another version of Table 3, where we have considered any real coefficient below the threshold $2.5\xb7{10}^{-3}$ to be equal to zero. Now the disagreement between theory and practice is still in place for small sample sizes (up to ${10}^{5}$), but it disappears when the sample size is $\u2a7e{10}^{6}$. In boldface are highlighted the coefficients for which the disagreement is still in place.

# | $\mathit{E}\left\{\mathit{X}\right\}$ (Sample Mean) | ${VAR}\left\{\mathit{X}\right\}$ (Sample Variance) |
---|---|---|

${10}^{3}$ | ${e}^{(\mathbf{0}.\mathbf{0260943}+9.55455{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(1.10438+19.1091{\eta}^{2}){\alpha}^{2}}-{e}^{(\mathbf{0}.\mathbf{0521886}+19.1091{\eta}^{2}){\alpha}^{2}}$ |

${10}^{4}$ | ${e}^{(-\mathbf{0}.\mathbf{0126293}+\mathbf{0}.\mathbf{0190467}\eta +9.56013{\eta}^{2}){\alpha}^{2}}$ | ${e}^{(0.949483+\mathbf{0}.\mathbf{0380934}\eta +19.1203{\eta}^{2}){\alpha}^{2}}-{e}^{(-\mathbf{0}.\mathbf{0252587}+\mathbf{0}.\mathbf{0380934}\eta +19.1203{\eta}^{2}){\alpha}^{2}}$ |

${10}^{5}$ | ${e}^{(-\mathbf{0}.\mathbf{00351095}+9.54501\eta )\alpha}$ | ${e}^{(1.00631-\mathbf{0}.\mathbf{0070219}\eta +19.09{\eta}^{2}){\alpha}^{2}}-{e}^{(\mathbf{0}.\mathbf{00315714}-\mathbf{0}.\mathbf{0070219}\eta +19.09{\eta}^{2}){\alpha}^{2}}$ |

${10}^{6}$ | ${e}^{9.54617}$ | ${e}^{(0.995845+19.0923{\eta}^{2}){\alpha}^{2}}-{e}^{19.0923}$ |

${10}^{7}$ | ${e}^{9.54666}$ | ${e}^{(1.00065+19.0933{\eta}^{2}){\alpha}^{2}}-{e}^{19.0933}$ |

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**MDPI and ACS Style**

Cococcioni, M.; Fiorini, F.; Pagano, M.
Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a *Numerically Verifiable* Infinite Variance. *Mathematics* **2023**, *11*, 1758.
https://doi.org/10.3390/math11071758

**AMA Style**

Cococcioni M, Fiorini F, Pagano M.
Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a *Numerically Verifiable* Infinite Variance. *Mathematics*. 2023; 11(7):1758.
https://doi.org/10.3390/math11071758

**Chicago/Turabian Style**

Cococcioni, Marco, Francesco Fiorini, and Michele Pagano.
2023. "Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a *Numerically Verifiable* Infinite Variance" *Mathematics* 11, no. 7: 1758.
https://doi.org/10.3390/math11071758