arXiv:2101.00516v1 [math.PR] 2 Jan 2021 akbeareetwt xeietldt,se[,7 ,11, 3, 7, [6, see data, experimental with agreement markable encountere have (q-distributions) distributions Tsallis accentuated. are regions p to h parameter The ae ymn uhr [6 ,1,1].Teq-distributions The m and 17]). chaos of f 10, characterization 9, the for ([16, physics statistical authors many by gated .Itouto n Preliminaries and Introduction 1. distribu Gaussian Keywords: q-independent dis of q-Gaussian sum the of of distribution transform paramet the q-Laplace kurtosis q-mome the the the of explicitely determine as expression we give Firstly, regarded we 1. Especially, be = may tion. q q-moments for new case q- classical these to the that related show results we some giv and Moreover, are established, uno q-moments are of and them properties normalized between Some called details. types in q-moments introduced c two when paper, problem not this and a In distribution is normalized corresponding there a popular, with value is generalizing distribution distribution q-normal probability a a is q-Gaussian The Abstract q for : ∗ r ipeoeprmtrtasomto fa rgnlden original an of transformation parameter one simple a are eea -nlge fcranpoaiiydistributions probability certain of q-analogues Several mi address: Email a a orsodn author Corresponding aoaoyo rbblt n ttsis aut fScienc of Faculty , and Probability of Laboratory > q 1 -alc rnfr;qGusa itiuin q-moments distribution; q-Gaussian transform; q-Laplace h oesnua ein r mlfid hl for while amplified, are regions singular more the , oepoete fqGusa distributions q-Gaussian of properties Some q [email protected] eae samcocp o xlrn ieetrgoso th of regions different exploring for microscope a as behaves f e aa Nahla Salah Ben q ( x = ) BnSlhNahla Salah (Ben Z 1 f f ( ( x x ) ) q q dx a ag ucs eas fterre- their of because success large a d sSa,Uiest fSa.Tunisia Sfax. of University Sfax, es a lircas hs distributions These ultifractals. ) r.Scnl,w opt the compute we Secondly, ers. h asinoe nsieof spite In one. Gaussian the -omldsrbto itself. distribution q-normal a tions. n n eea relationships several and en, rbto.Fnly estudy we Finally, tribution. oet r loobtained. also are moments aebe eetyinvesti- recently been have 2,adrfrne therein. references and 12], t fqGusa distribu- q-Gaussian of nts luaiga expectation an alculating iyfunction sity aebe nrdcdin introduced been have mlzdqmmnsare q-moments rmalized < q eeeaainof generelazation a 1 q-estimator. ; h essingular less the f measure e according In particular, the q-Gaussian distribution is also well-known as a gneralisation of the Gaussian, or the . This distribution can also represent the heavy tailed distribution such as the Student-distribution or the distribution with bounded support such as the semicircle of Wigner. For these reasons, the q-Gaussian distribu- tion has been applied in the fields of statistical mechanics, geology, finance, and machine learning. Admitting the q-normal distribution is in demand as above, there exists a prob- lem to calculate the expectation value with a corresponding q-distribution not a q-normal distribution itself. But we have an amazing property such that an escort distribution ob- tained by a q-normal distribution with a parameter q and a is another q-normal distribution with a different value of q and a scaled variance. Then calculating an ex- pectation value with an escort distribution corresponds to calculating the expectation value with another associated q-normal distribution, but it gets even the question why an expectation value should be calculated by another q-normal distribution. We call the procedure to get another q-normal distribution from a given q-normal distribution through an escort distribution proportion. Furthermore, we target attention on q-Gaussians, an essential tool of q-statistics [14], that was not discussed in [2]. The q-Gaussian behavior is often detected in quite distinct settings [14]. It is well known that, in the literature, there are two types of q-Laplace transforms, and they are studied in detail by several authors ([20, 15], etc.). Recently Tsallis et al. have been interested in calculating the of q-Gaussian and have proved a generalization of the central theorem for 1 q < 3. The case q < 1 ≤ requires essentially different technique, therefore we leave it for a separate paper. In this paper, we propose new definitions of the q-laplace transform of some probability distributions. These results are motivated by recent developments in the calculation of Fourier transforms, where new formulas have been defined [17]. In this article, we develop our results into four sections. In Section 2, we recall some known definitions and notations from the q-theory. In Section 3, we give definitions of some q-analogues of mean and variance. In Section 4, we introduce the q-Gaussian distribution includes some properties. In Section 5, we give the news formula of Laplace transform and we treat kurtosis both in its standard definition and in q statistics, namely q-kurtosis. In Section 6, we estimate the q-mean and q-variance. We start with definitions and facts from the q-calculus.

2. q-theory calculus

Assume that q be a fixed number satisfying q [0, 1]. If is a classical object,say, 1 qn ∈ its q-version is defined by [x]q = − . As is well know, the q-exponential and the 1 q − q-, which are denoted by eq(x) and lnq(x), are respectively defined as eq(x)=

2 1 − 1−q x1 q −1 x⊗qy x y [1+(1 q)x]+ and lnq(x)= 1−q , (x> 0). For q-exponential, the relations eq = eq eq x−+y x y and e = e q e hold true. These relations can be rewritten equivalently as follows: q q ⊗ q lnq(x q y) = lnq(x) + lnq(y), and lnq(xy) = lnq(x) q lnq(y). ⊗ ⊗ A q-algebra can also be defined in [16] by applying the generalized operation for sum and product: x q y = x + y + (1 q)xy, ⊕ − 1−q 1−q 1−q x q y = [x + y 1] , ⊗ − + with the following neutral and inverse elements:

−1 x q (x)q =0, with (xq)= x[1 + (1 q)x] ⊗ −

1 −1 −1 1−q 1−q x q (x )q =1, with :(x )q = [2 x ] . ⊗ − + For the new algebraic operation, q-exponential and q-logarithm have the following properties:

x y x⊗qy Properties 2.1. 1. eq eq = eq x y x+y 2. e q e = e q ⊗ q q 3. log (xy) = log (x) q log (y) q q ⊗ q 4. log (x q y) = log (x) + log (y) q ⊗ q q

It can be easily proved that the operation q and q satisfy commutativity and ⊗ ⊕ associativity. For the operator q, the identity additive is 0, while for the operator q the identity multiplicative is 1 [1].⊕ Two distinct mathematical tools appears in the study⊗ of physical phenomena in the complex media which is characterized by singularities in a compact space. From the associativity of q and q, we have the following formula : ⊕ ⊗

1 n t q t q .... q t = [1 + (1 q)t] 1 ⊕ ⊕ ⊕ 1 q { − − } − 1 ⊗q n 1−q 1−q t = t q t q ... q t = nt (n 1) . ⊗ ⊗ ⊗ − − The real space vector with regular sum and product operations R(+, ) is a field, and × the R( q, q) defines a quasi-field. ⊕ ⊗

3 3. q-mean and q-variance values Let q be a real number and f be a properly normalized probability density with suppf R of some X such that the quantity ⊆ +∞ f(x)dx =1. Z−∞ The mean m is defined, of a given X, as follows

+∞ E(X)= m = xf(x)dx. Z−∞

The variance V is defined, of a given X, as follows

∞ V (X)= (x m)2f(x)dx. Z−∞ −

The unnormalized q-moments , of a given X, is defined as

+∞ q Eq(X)= mq = x[f(x)] dx. Z−∞

2 Similarly, the unnormalized q-variance, σ2q−1 is defined analogously to the usual second order central moment, as

+∞ 2 2 2q−1 V2q−1(X)= σ2q−1 = (x m2q−1) [f(x)] dx. Z−∞ −

On the other hand, we denote by fq(x) the normalized density (see e.g. [13]) and defined as [f(x)]q fq(x)= . νq(f) where +∞ q νq(f)= [f(x)] dx < . Z−∞ ∞ The normalized q-mean values, of a given X, is

+∞ Eq(X)= mq = xfq(x)dx Z−∞ The normalized q-variance values, of a given X, is

+∞ 2 2 V 2q−1(X)= σ2q−1 = (x m2q−1) f2q−1(x)dx. Z−∞ −

4 4. q-Gaussian distribution In this section, we review the q-Gaussian distribution, or the q-normal distribution according to Furuichi [16] and Suyari [8]. Let β be a positive number. We call the 2 2 q-Gaussian Nq(m, σ ) with parameters m and σ > 0 if its density is defined by β(x m)2 − − √β σ2 f(x)= eq , x R σCq ∈ with q < 3, q =1; and Cq is the normalizing constant, namely 6 1 1 2 3 q 1 B − , 1

√β −βy2 Nq(0, 1)(y)= eq . Cq

Note that, if 2 Y Nq(0, 1) then X = m + σY Nq(m, σ ). (1) ∼ ∼ If we change the value of q, we can represent various types of distributions. The q- Gaussian distribution represents the usual Gaussian distribution when q = 1, has com- pact support for q < 1, and turns asymptotically as a power law for 1 q < 3. For 3 q, the form given is not normalizable. The usual variance (second order≤ moment) is ≤ 5 3 q finite for q < , and, for the standard q-Gaussian Nq(0, 1), is given by V (Y )= − . 3 5 3q The usual variance of the q-Gaussian diverges for 5 q < 3 , however the q-variance− 3 ≤ remains finite for the full range

5 5. New q-Laplace transforms

From now, we assume that 1 q < 3. For these values of q we introduce the ≤ q-Laplace transform Lq as an operator, which coincides with the Laplace transform if q =1. Note that the q-Laplace transform is defined on the basis of the q-product and the q-exponential, and, in contrast to the usual Laplace transform, is a nonlinear transform for q (1, 3). The∈ q-Laplace transform of a random variable X with density function f is defined by the formula

θx Lq(X)(θ)= eq q f(x)dx, . Zsuppf ⊗ where the is understood in the Lebesgue sense. The following lemma establishes the expression of the q-Laplace transform in terms of the standard product, instead of the q-product.

Lemma 5.1. The q-Laplace transform of a random variable X with density f is ex- pressed as ∞ θx(f(x))q−1 Lq(X)(θ)= f(x)eq dx. (3) Z−∞

Proof. For x suppf, we have ∈ 1 1 iθx 1−q − q−1 − e q f(x) = [1 + (1 q)θx +(f(x)) 1] 1 q = f(x)[1 + (1 q)θx(f(x)) ] 1 q (4) q ⊗ − − − Integrating both sides of Eq. (4) we obtain (3). Let X be a random variable defined on the probability space (Ω,F,P ) with density function f Lq. It can be verified that the of the q-Laplace transform ∈ Lq(X)(θ) are closely related to an appropriate set of unnormalized q-moments of the original probability density. Assume that Lq(X)(θ) < + in a neighbor of 0. Indeed, the first few low-order derivatives (including∞ the zeroth order) are given by

Lq(X)(0) = 1

∞ ∂Lq(X)(θ) q = x(f(x)) dx = Eq(X) ∂θ θ=0 Z−∞

2 2 ∞ ∂ Lq(X)(θ ) 2 2q−1 2 = q x (f(x)) dx = qE2q−1(X ) ∂θ θ=0 Z−∞

3 3 ∞ ∂ Lq(X)(θ ) 3 3q−2 3 = q(2q 1) x (f(x)) dx = q(2q 1)E3q−2(X ). ∂θ θ=0 − Z−∞ −

6 4 4 ∞ ∂ Lq(X)(θ ) 4 4q−3 4 = q(2q 1)(3q 2) x (f(x)) dx = q(2q 1)(3q 2)E4q−3(X ). ∂θ θ=0 − − Z−∞ − −

The general n- is

n− ∂nL (X)(θn) 1 ∞ q = (1 + m(q 1)) xn(f(x))1+n(q−1)dx, n =1, 2, 3.... ∂θ θ=0 Y − Z−∞ m=0

Note that, in the case n =1 the first derivative of the Laplace transform corresponds to Eq(X).

Proposition 5.2. Let 1 6 q < 3 and let X be a random variable following a q-Gaussian 2 3 q 2 5 distribution Nq(m, σ ) then E(X)= m and V (X)= − σ with 1 q < . 5 3q ≤ 3 − Proof. 1. The first moment, of a given X, is

β(x m)2 ∞ − − √β σ2 E(X)= xeq dx σCq Z−∞ ∞ √β −βy2 = (σy + m)eq dy Cq Z−∞ ∞ ∞ √β −βy2 √β −βy2 = σ yeq dy + m eq dy = m Cq Z−∞ Cq Z−∞

2. The second order moment of the standard Gaussian Nq(0, 1) is computed as

∞ 2 √β 2 −βy2 E(Y )= y eq dy Cq Z−∞ ∞ 1 −βy2 2−q = (eq ) dy 2√β(2 q)Cq Z−∞ − ∞ 1 2 1 e−β(2−q)y dy,q = q1 1 = 2√β(2 q)Cq Z−∞ 2 q − − 2 2 1 The substitution β1z = β(2 q)y , β1 = − 3 q1 ∞ − √β 2 1 −β1z = 3 eq1 dz 2β(2 q) 2 Cq Z−∞ − ∞ Cq √β 2 1 1 e−β1z dz = 3 q1 2β(2 q) 2 Cq Z−∞ Cq1 − 1 Cq1 = 3 2β (2 q) 2 Cq 1 −

7 6 6 5 The condition 1 q < 3 implies that 1 q1 < 3 . By using the identity B(x+1,y)= x x+y B(x, y) we obtain the ratio between Cq and Cq1 as 3

Cq 2(2 q)2 1 = − (5) Cq 5 3q − By applying the formula V (X)= E(X2) (E(X))2, we obtain the result − In this theorem, we give the average of the fourth power of the standardized deviations from the q-mean. In this theorem, we determine the q-kurtosis of q- Gaussian distribution (Nq(0, 1)).

Theorem 5.3. Let 1 6 q < 3 and let X be a random variable following a q-central Gaussian distribution Nq(0, 1), then the coefficient of kurtosis is E(Y 4) 3(5 3q) 7 Kurt[Y ]= = − , 1 q < (E(Y 2))2 (7 5q) ≤ 5 − Proof. 1. The fourth central moment moment of the standard Gaussian Nq(0, 1) is computed as

∞ 4 √β 4 −βy2 E(Y )= y eq dy Cq Z−∞ ∞ 3 2 −βy2 2−q = y (eq ) dy 2√β(2 q)Cq(1) Z−∞ − 2 2 The substitution β1z = β(2 q)y ∞ − β C √β 2 3 1 q 1 2 −β1z 1 5 = 5 z eq dz with 1 q1 = < 2 1 2β (2 q) 2 Cq Z−∞ Cq1 ≤ 2 q 3 − − 3β1 Cq1 2 = 5 Eq1 (Z ) 2β2(2 q) 2 Cq − Using equation 5, we obtain 3(3 q)2 7 = − , 1 q < (5 3q)(7 5q) ≤ 5 − − According to Proposition 5.2, we obtain the result.

2 For 1 q < 7 a value greater than 3(3−q) indicates a leptokurtic distribution; a ≤ 5 (5−3q)(7−5q) 3(3−q)2 values less than (5−3q)(7−5q) indicates a platykurtic distribution. For the sample estimate 3(3−q)2 X, (5−3q)(7−5q) is subtracted so that a positive value indicates leptokurtosis and a negative value indicates platykurtosis.

8 Theorem 5.4. Let 1 6 q < 3 and let X be a random variable following a q-Gaussian 2 distribution Nq(m, σ ), then θ2a2q−2σ2 q−1 θma − 3−q 4β 2 √β 1. Lq(X)(θ)= eq , with a = and θ R   σCq ∈ 3 q − ∞ m(3 q) 2 E X x f x qdx 2. q( )= ( ( )) = − q−1 Z−∞ 2(σCq) ∞ 2 2 2q−1 1 2 2 E − X x f x dx q σ q m 3. 2q 1( )= ( ( )) = q−2 2q−2 [(3 ) +( + 1) ] Z−∞ 4q(3 q) (σCq) − −

9 √β Proof. 1. From definition of q-Laplace transform, it following by denote a = σCq

β(x m)2 ∞ − − θx σ2 Lq(X)(θ)= eq q aeq dx Z−∞ ⊗ (by appliying lemma 5.1) β(x m)2 2 q−1 β(x m) − −2 ∞ q−1 − − θxa eq σ σ2 = a eq eq   dx Z−∞ β(x m)2 ∞ − − θxaq−1 σ2 = a eq q eq dx Z−∞ ⊗ 2 β(x m) − ∞ − − + θxaq 1 σ2 = a eq dx Z−∞ 2 2 q−1 β(x m) β σ θa 2 q−1 ∞ − − + ( ) + θa m σ2 σ2 2β = a eq dx Z−∞ 2 q−1 β σ θa 2 q−1 2 q−1 2 ( ) + θa m β σ θa 2 q−1 β(x m) 2 σ 2β q−1 ∞ ( ) + θa m − − (eq ) σ2 2β σ2 = a eq eq dx Z−∞ 2 q−1 2 β σ θa 2 q−1 βγ(x m) ( ) + θa m ∞ − − σ2 2β σ2 = aeq eq dx Z−∞ β σ2θaq−1 ( )2 + θaq−1m Cq σ2 2β = σ a eq 1 √β β σ2θaq−1 ( )2 + θaq−1m σ2 2β eq = 2 q−1 β σ θa 2 q−1 v (q−1)( ( ) + θa m) u σ2 2β ue t 2

2 q−1 2 ( β σ θa 2 q−1 q−1 2 σ where γ = eq ( ) + θa m) andσ = . σ2 2β 1 γ

10 Hence, applying again Lemma 5.1, we have

3−q θ2a2q−2σ2 2 θmaq−1−  4β  Lq(X)(θ)= eq θm(3 q) θ2a2q−2σ2(3 q) − aq−1 − 2 − 8β = eq1

2. We compute the first and the second derivative of Lq(X), with respect to θ, we obtain ′ Eq(X)= Lq(X)(0)

2 1 E − (X )= L” (0) 2q 1 q q Intending to interpret these moments, we consider a wonderful property such that an escort distribution obtained by a q-normal distribution with variance σ2 is equivalent to 3 q another q-normal distribution with q =2 1 and a variance − σ2 with 1 q < 3. 1 − q q +1 ≤ 2 Proposition 5.5. Let X be a random variable following a q-Gaussian distribution Nq(m, σ ),then +∞ [f(x)]q 1. Eq(X)= x dx = m Z−∞ νq(f) +∞ 2q−1 2 2 [f(x)] 3 q 2 2. V 2q−1(X)= σ2q−1 = (x m) dx = − σ Z−∞ − ν2q−1(f) q +1 Proof. 1. Let’s begin with observing a following proportion on a given q-Normal distribution: 2q 1 β 3 q q = − , σ2 = 3 σ2 = − σ2 3 q 3 (q)β q +1 Under this relations, we have β(x m)2 − 2 −2 q β(x m) qβ σ − eq −2 √β σ β3   eq3 σCq νq ∝ 2 Nq (m, σ ) ∝ 3 3

Therefore, +∞ q +∞ [f(x)] 2 Eq(X)= x dx = xNq3 (m, σ3)(x)dx Z−∞ νq(f) Z−∞ By applying proposition 5.2, we obtain the result.

11 2 2. The escort function is proportional to the q-gaussian Nq4 (m, σ ) 3q 2 β 3 q q = − , σ2 = 4 σ2 = − σ2. 4 2q 1 4 (2q 1)β 3q 1 − − − Hence β(x m)2 − −2 q σ 2 √β eq − (x−m) (2q−1)β σ2 β4 (6)   eq4 σCq ν2q−1 ∝ 2 Nq (m, σ ) ∝ 4 4 Then

+∞ 2q−1 +∞ 2 [f(x)] 2 2 3 q4 2 V 2q−1(X)= (x m) dx = (x m) Nq4 (m, σ4)(x)dx = − σ4 Z−∞ − ν q− (f) Z−∞ − 5 3q 2 1 − 4 By applying proposition 5.2we obtain the result.

3 3(q+1)2 In rhis theorem, we prove that for q < 5 a value greater than (5q−3)(3q−1) indicates a 3(q+1)2 leptokurtic distribution; a values less than (5q−3)(3q−1) indicates a platykurtic distribution. 3(q+1)2 For the sample estimate X, (5q−3)(3q−1) is subtracted so that a positive value indicates leptokurtosis and a negative value indicates platykurtosis.

Theorem 5.6. Let Y be a random variable following a q-Central Gaussian distribution Nq(0, 1) then the coefficient of normalized kurtosis is

E(Y 4) 3(q + 1)2 3 Kurt[Y ]= = , 1 q < (E(Y 2))2 (5q 3)(3q 1) ≤ 5 − − Proof. Checking a following proportion on a given q-Normal distribution: 5q 4 β q = − , σ2 = 1 . 1 4q 3 1 (4q 3)β − −

βx2 − 2 (4q−3)β −x β1 √β 4q−3 ν4q−3 β1 ( ) eq eq1 σCq ∝ 2 Nq (0, σ ) ∝ 1 1

12 βz2 +∞ − +∞ 4 4 √β 4q−3 ν4q−3 4 2 Then E(Z )= z ( ) eq = z Nq1 (0, σ1)(z)dz. Z−∞ σCq Z−∞ 2 4 3(3 q1) 4 2 Hence, according to Theorem 5.3, we obtain E(Z )= − σ1; Z Nq1 (0, σ1). (5 3q1)(7 5q1) ∼ we obtain − − 3(3 q)2 E(Z4)= − . (5q 3)(3q 1) − − Furthermore by observing a following proportion on a given q-Normal distribution: 3q 2 β q = − , σ2 = 2 2 2q 1 2 (2q 1)β − − βx2 − 2 (2q−1)β √β −x β1 2q−1 ν2q−1 β2 ( ) eq eq2 (7) σCq ∝ 2 Nq (0, σ ) ∝ 2 2 Then βz2 +∞ − +∞ 2 2 √β 2q−1 ν2q−1 2 2 E(Z )= z ( ) eq = z Nq2 (0, σ1)(z)dz. Z−∞ σCq Z−∞

Hence, according to proposition 5.2, we obtain

2 2 (3 q2)σ2 2 E(Z )= − ; Z Nq (0, σ ) (5 3q ) ∼ 2 2 − 2 3 q E(Z2)= − . q +1 q-estimator for random variables are arising from non-extensive statistical mechanics. In this section, we will estimate the q-mean and q-variance using the notions of q-Laplace transform, q-independence.

6. Estimator of q-Mean and q-variance

Definition 6.1. Two random variables X1 and X2 are said to be q-independent if

Lq(X + X )(θ)= Lq(X )(θ) q Lq(X )(θ). 1 2 1 ⊗ 2

13 Definition 6.2. Let Xn be a sequence of identically distributed random variables and n m = E(X1). Denote Sn = Xk.. By definition Xk,k = 1, 2, 3, ..., is said to be q- Xk=1 independent of the first type (or q-i.i.d.) if for all n =2, 3, 4, ..., the relations

Lq[Sn nm](θ)= Lq[X m](θ) q .... q Lq[Xn m](θ) − 1 − ⊗ ⊗ − hold.

Proposition 6.3. Let X1 and X2 be tow q-independent random variables following re- 2 2 spectively Nq(m1, σ1 ) and Nq(m2, σ2 ). Then y 2 2 X1 + X2 Nq(m1 + m2, σ1 + σ2 ) Proof.

Lq(X + X )(θ)= Lq(X )(θ) q Lq(X )(θ) 1 2 1 ⊗ 2 2 2q−2 3 q 2 2 θ a 3 q 2 2 q−1 − (σ + σ ) + − (σ + σ )θa 2 1 2 4β 2 1 2 = eq1 θ2a2q−2 (σ2+σ2) +(σ2+σ2)θaq−1 3 q 1 2 4β 1 2 − = eq 2  

Note that if X1 and X2 are q-Gaussian and q-independent random variables with distri- 2 2 butions Nq(m1, σ1 ) and Nq(m2, σ2) respectively then 3 q 5 V (X + X )= − (σ2 + σ2);1 6 q < 1 2 5 3q 1 2 3 − and 3 q 2 2 V q− (X + X )= − (σ + σ )= V q− (X )+ V q− (X ); 2 1 1 2 q +1 1 2 2 1 1 2 1 2

In this case cov(X1,X2)=0, because V (X1 + X2)= V (X1)+ V (X2)+2cov(X1,X2).

Corollary 6.4. Let X1,X2, ..., Xn be n q-independent random variables with same q- n 2 1 σ2 Gaussian distribution Nq(m, σ ), then Xn = n Xi follows the q-Gaussian Nq(m, n ). Xi=1

Observe that V (X) 0 as n . Since E(X) = m, then the estimates of m becomes increasingly−→ concentrated−→ around ∞ the true population parameter. Such an

14 estimate is said to be consistent. 2 1 n 2 2 The empirical variance Sn = (Xi Xn) is not an unbiased estimate of σ . Indeed n i=1 − P 2 n 1 E(Sn )= − V (X ) n 1 n 1 3 q = − − σ2 n 5 3q − Therefore n 5 3q σ2 = − S2. n 1 3 q n − − is an unbiased estimate of σ2. b

Proposition 6.5. Law of Large Numbers (LLN): If the distribution of the i.i.d. q- independent X , ..., Xn is such that X has finite q-expectation, i.e. Eq(X ) < , then 1 1 | 1 | ∞ the sample average X1 + ... + Xn Xn = Eq(X ) n −→ 1 converges to its expectation in probability.

Theorem 6.6. (CLT):[18, 10, 19, 4] For q (1, 2), if X ,X , ..Xn ∈ 1 2 are q-independent and identically distributed with q-mean mq and a finite second (2q 1)- 2 − moment σ2q−1, then X1 + ... + Xn nmq Zn = − Cq,n,σ q−1 converges to Nq−1 (0, 1) Gaussian distribution.

3 q Let X be an arbitrary random variable with known variance − σ2, and let X ,X , ..., X 5 3q 1 2 n − 2 be n- q-independent random variables with commun Gaussian distribution Nq(m, σ ). According to central limit theorem, the confidence interval for m with level 1 α for arbitrary data and known σ2. is defined as −

z − α σ 1 2 m (Xn . ∈ ± Cq,N √n

α Where z1− α is the quantile for Nq(0, 1) with level 1 : 2 − 2 α Nq(0, 1)(x)dx =1 α,α ]0, 1[. Z−α − ∈

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