aa r X i v : . [ m a t h . P R ] J a n Some properties of q-Gaussian distributions
Ben Salah Nahla a a Laboratory of Probability and Statistics, Faculty of Sciences Sfax, University of Sfax. Tunisia
Abstract
The q-Gaussian is a probability distribution generalizing the Gaussian one. In spite ofa q-normal distribution is popular, there is a problem when calculating an expectationvalue with a corresponding normalized distribution and not a q-normal distribution itself.In this paper, two q-moments types called normalized and unormalized q-moments areintroduced in details. Some properties of q-moments are given, and several relationshipsbetween them are established, and some results related to q-moments are also obtained.Moreover, we show that these new q-moments may be regarded as a generelazation ofthe classical case for q = 1. Firstly, we determine the q-moments of q-Gaussian distribu-tion. Especially, we give explicitely the kurtosis parameters. Secondly, we compute theexpression of the q-Laplace transform of the q-Gaussian distribution. Finally, we studythe distribution of sum of q-independent Gaussian distributions.
Keywords: q-Laplace transform; q-Gaussian distribution; q-moments; q-estimator.
1. Introduction and Preliminaries
Several q-analogues of certain probability distributions have been recently investi-gated by many authors ([16, 9, 10, 17]). The q-distributions have been introduced instatistical physics for the characterization of chaos and multifractals. These distributions f q are a simple one parameter transformation of an original density function f accordingto f q ( x ) = f ( x ) q Z f ( x ) q dx The parameter q behaves as a microscope for exploring different regions of the measure p : for q > , the more singular regions are amplified, while for q < the less singularregions are accentuated.Tsallis distributions (q-distributions) have encountered a large success because of their re-markable agreement with experimental data, see [6, 7, 3, 11, 12], and references therein. ∗ a Corresponding author
Email address: [email protected] (Ben Salah Nahla a )
1n particular, the q-Gaussian distribution is also well-known as a gneralisation of theGaussian, or the Normal distribution. This distribution can also represent the heavytailed distribution such as the Student-distribution or the distribution with boundedsupport 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 machinelearning. 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-normaldistribution 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 variance is another q-normaldistribution with a different value of q and a scaled variance. Then calculating an ex-pectation value with an escort distribution corresponds to calculating the expectationvalue with another associated q-normal distribution, but it gets even the question whyan expectation value should be calculated by another q-normal distribution. We callthe procedure to get another q-normal distribution from a given q-normal distributionthrough 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 distinctsettings [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 etal. have been interested in calculating the Fourier transform of q-Gaussian and haveproved a generalization of the central limit theorem for ≤ q < . The case q < requires essentially different technique, therefore we leave it for a separate paper. Inthis paper, we propose new definitions of the q-laplace transform of some probabilitydistributions. These results are motivated by recent developments in the calculation ofFourier transforms, where new formulas have been defined [17].In this article, we develop our results into four sections. In Section 2, we recall someknown definitions and notations from the q-theory.In Section 3, we give definitions of some q-analogues of mean and variance. In Section4, we introduce the q-Gaussian distribution includes some properties. In Section 5, wegive the news formula of Laplace transform and we treat kurtosis both in its standarddefinition and in q statistics, namely q-kurtosis. In Section 6, we estimate the q-meanand 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 , . If is a classical object,say,its q-version is defined by [ x ] q = 1 − q n − q . As is well know, the q-exponential and theq-logarithm, which are denoted by e q ( x ) and ln q ( x ) , are respectively defined as e q ( x ) = − q ) x ] − q + and ln q ( x ) = x − q − − q , ( x > . For q-exponential, the relations e x ⊗ q yq = e xq e yq and e x + yq = e xq ⊗ q e yq hold true. These relations can be rewritten equivalently as follows: ln q ( x ⊗ q y ) = ln q ( x ) + ln q ( y ) , and ln q ( xy ) = ln q ( x ) ⊗ q ln q ( y ) . A q-algebra can also be defined in [16] by applying the generalized operation for sumand product: x ⊕ q y = x + y + (1 − q ) xy,x ⊗ q y = [ x − q + y − q − − q + , with the following neutral and inverse elements: x ⊗ q ( x ) q = 0 , with ( x q ) = x [1 + (1 − q ) x ] − x ⊗ q ( x − ) q = 1 , with : ( x − ) q = [2 − x − q ] − q + . For the new algebraic operation, q-exponential and q-logarithm have the followingproperties:
Properties 2.1. e xq e yq = e x ⊗ q yq e xq ⊗ q e yq = e x + yq log q ( xy ) = log q ( x ) ⊗ q log q ( y ) log q ( x ⊗ q y ) = log q ( x ) + log q ( y ) It can be easily proved that the operation ⊗ q and ⊕ q satisfy commutativity andassociativity. 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 studyof physical phenomena in the complex media which is characterized by singularities in acompact space.From the associativity of ⊕ q and ⊗ q , we have the following formula : t ⊕ q t ⊕ q .... ⊕ q t = 11 − q { [1 + (1 − q ) t ] n − } t ⊗ q n = t ⊗ q t ⊗ q ... ⊗ q t = nt − q − ( n − − q . The real space vector with regular sum and product operations R (+ , × ) is a field, andthe R ( ⊕ q , ⊗ q ) defines a quasi-field. 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 random variable X such that the quantity Z + ∞−∞ f ( x ) dx = 1 . The mean m is defined, of a given X , as follows E ( X ) = m = Z + ∞−∞ xf ( x ) dx. The variance V is defined, of a given X , as follows V ( X ) = Z ∞−∞ ( x − m ) f ( x ) dx. The unnormalized q-moments , of a given X , is defined as E q ( X ) = m q = Z + ∞−∞ x [ f ( x )] q dx. Similarly, the unnormalized q-variance, σ q − is defined analogously to the usual secondorder central moment, as V q − ( X ) = σ q − = Z + ∞−∞ ( x − m q − ) [ f ( x )] q − dx. On the other hand, we denote by f q ( x ) the normalized density (see e.g. [13]) and definedas f q ( x ) = [ f ( x )] q ν q ( f ) . where ν q ( f ) = Z + ∞−∞ [ f ( x )] q dx < ∞ . The normalized q-mean values, of a given X , is E q ( X ) = m q = Z + ∞−∞ xf q ( x ) dx The normalized q -variance values, of a given X , is V q − ( X ) = σ q − = Z + ∞−∞ ( x − m q − ) f q − ( x ) dx. . q-Gaussian distribution In this section, we review the q-Gaussian distribution, or the q-normal distributionaccording to Furuichi [16] and Suyari [8]. Let β be a positive number. We call theq-Gaussian N q ( m, σ ) with parameters m and σ > if its density function is defined by f ( x ) = √ βσC q e − β ( x − m ) σ q , x ∈ R with q < , q = 1 ; and C q is the normalizing constant, namely C q = Z ∞−∞ e − x q dx = (cid:16) q − (cid:17) B (cid:16) − q q − , (cid:17) < q < √ π, q = 1 (cid:16) − q (cid:17) B (cid:16) − q − q , (cid:17) − ∞ < q < and B ( a, b ) denotes the beta function. The widht parameter of the distribution is char-acterized by β = 13 − q . Denote a general q-Gaussian random variable X with parameters m and σ as X ∼ N q ( m, σ ) , and call the special case of m = 0 and σ = 1 a standard q-Gaussian Y ∼ N q (0 , . The density of the satndard q-Gaussian distribution may then be written as N q (0 , y ) = √ βC q e − βy q . Note that, if Y ∼ N q (0 , then X = m + σY ∼ N q ( 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 < , and turns asymptotically as a power law for ≤ q < . For ≤ q, the form given is not normalizable. The usual variance (second order moment) isfinite for q < , and, for the standard q-Gaussian N q (0 , , is given by V ( Y ) = 3 − q − q . The usual variance of the q-Gaussian diverges for ≤ q < , however the q-varianceremains finite for the full range ∞ < q < , equal to unity for the standard q-Gaussian.Finally, we can easily check that there are relationships between different values of q.For example, e − y q = ( e − qy − q ) q . (2)In this section, we consider the q-analogues of the Laplace transform, which we call theq-Laplace transform, and investigate some of its properties.5 . New q-Laplace transforms From now, we assume that ≤ q < . For these values of q we introduce theq-Laplace transform L q 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 theq-exponential, and, in contrast to the usual Laplace transform, is a nonlinear transformfor q ∈ (1 , . The q-Laplace transform of a random variable X with density function f is definedby the formula L q ( X )( θ ) = Z suppf e θxq ⊗ q f ( x ) dx, . where the integral is understood in the Lebesgue sense.The following lemma establishes the expression of the q-Laplace transform in termsof 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 L q ( X )( θ ) = Z ∞−∞ f ( x ) e θx ( f ( x )) q − q dx. (3) Proof.
For x ∈ suppf, we have e iθxq ⊗ q f ( x ) = [1 + (1 − q ) θx + ( f ( x )) − q − − q = f ( x )[1 + (1 − q ) θx ( f ( x )) q − ] − q (4)Integrating both sides of Eq. ( ) we obtain (3) . Let X be a random variable defined on the probability space (Ω , F, P ) with densityfunction f ∈ L q . It can be verified that the derivatives of the q-Laplace transform L q ( X )( θ ) are closely related to an appropriate set of unnormalized q-moments of theoriginal probability density. Assume that L q ( X )( θ ) < + ∞ in a neighbor of 0.Indeed, the first few low-order derivatives (including the zeroth order) are given by L q ( X )(0) = 1 ∂L q ( X )( θ ) ∂θ (cid:12)(cid:12)(cid:12) θ =0 = Z ∞−∞ x ( f ( x )) q dx = E q ( X ) ∂ L q ( X )( θ ) ∂θ (cid:12)(cid:12)(cid:12) θ =0 = q Z ∞−∞ x ( f ( x )) q − dx = qE q − ( X ) ∂ L q ( X )( θ ) ∂θ (cid:12)(cid:12)(cid:12) θ =0 = q (2 q − Z ∞−∞ x ( f ( x )) q − dx = q (2 q − E q − ( X ) . L q ( X )( θ ) ∂θ (cid:12)(cid:12)(cid:12) θ =0 = q (2 q − q − Z ∞−∞ x ( f ( x )) q − dx = q (2 q − q − E q − ( X ) . The general n-derivative is ∂ n L q ( X )( θ n ) ∂θ (cid:12)(cid:12)(cid:12) θ =0 = n − Y m =0 (1 + m ( q − Z ∞−∞ x n ( f ( x )) n ( q − dx, n = 1 , , .... Note that, in the case n = 1 the first derivative of the Laplace transform correspondsto E q ( X ) . Proposition 5.2.
Let q < and let X be a random variable following a q-Gaussiandistribution N q ( m, σ ) then E ( X ) = m and V ( X ) = 3 − q − q σ with ≤ q < . Proof.
1. The first moment, of a given X, is E ( X ) = √ βσC q Z ∞−∞ xe − β ( x − m ) σ q dx = √ βC q Z ∞−∞ ( σy + m ) e − βy q dy = σ √ βC q Z ∞−∞ ye − βy q dy + m √ βC q Z ∞−∞ e − βy q dy = m
2. The second order moment of the standard Gaussian N q (0 , is computed as E ( Y ) = √ βC q Z ∞−∞ y e − βy q dy = 12 √ β (2 − q ) C q Z ∞−∞ ( e − βy q ) − q dy = 12 √ β (2 − q ) C q Z ∞−∞ e − β (2 − q ) y q dy, q = 12 − q The substitution β z = β (2 − q ) y , β = 13 − q = √ β β (2 − q ) C q Z ∞−∞ e − β z q dz = C q β (2 − q ) C q Z ∞−∞ √ β C q e − β z q dz = 12 β (2 − q ) C q C q q < implies that q < . By using the identity B ( x +1 , y ) = xx + y B ( x, y ) we obtain the ratio between C q and C q as C q C q = 2(2 − q ) 325 − q (5)By applying the formula V ( X ) = E ( X ) − ( E ( X )) , we obtain the resultIn this theorem, we give the average of the fourth power of the standardized deviationsfrom the q-mean.In this theorem, we determine the q-kurtosis of q- Gaussian distribution ( N q (0 , ). Theorem 5.3.
Let q < and let X be a random variable following a q-centralGaussian distribution N q (0 , , then the coefficient of kurtosis is Kurt [ Y ] = E ( Y )( E ( Y )) = 3(5 − q )(7 − q ) , ≤ q < Proof.
1. The fourth central moment moment of the standard Gaussian N q (0 , iscomputed as E ( Y ) = √ βC q Z ∞−∞ y e − βy q dy = 32 √ β (2 − q ) C q (1) Z ∞−∞ y ( e − βy q ) − q dy The substitution β z = β (2 − q ) y = 3 β β (2 − q ) C q C q Z ∞−∞ √ β C q z e − β z q dz with ≤ q = 12 − q <
53= 3 β β (2 − q ) C q C q E q ( Z ) Using equation 5, we obtain = 3(3 − q ) (5 − q )(7 − q ) , ≤ q < According to Proposition . , we obtain the result.For ≤ q < a value greater than − q ) (5 − q )(7 − q ) indicates a leptokurtic distribution; avalues less than − q ) (5 − q )(7 − q ) indicates a platykurtic distribution. For the sample estimate X , − q ) (5 − q )(7 − q ) is subtracted so that a positive value indicates leptokurtosis and a negativevalue indicates platykurtosis. 8 heorem 5.4. Let q < and let X be a random variable following a q-Gaussiandistribution N q ( m, σ ) , then L q ( X )( θ ) = (cid:16) e θma q − − θ a q − σ β q (cid:17) − q , with a = √ βσC q and θ ∈ R E q ( X ) = Z ∞−∞ x ( f ( x )) q dx = m (3 − q ) 3 − q σC q ) q − E q − ( X ) = Z ∞−∞ x ( f ( x )) q − dx = 14 q (3 − q ) q − ( σC q ) q − [(3 − q ) σ + ( q + 1) m ] roof.
1. From definition of q-Laplace transform, it following by denote a = √ βσC q L q ( X )( θ ) = Z ∞−∞ e θxq ⊗ q ae − β ( x − m ) σ q dx (by appliying lemma 5.1) = a Z ∞−∞ e − β ( x − m ) σ q e θxa q − (cid:16) e − β ( x − m ) σ q (cid:17) q − q dx = a Z ∞−∞ e θxa q − q ⊗ q e − β ( x − m ) σ q dx = a Z ∞−∞ e − β ( x − m ) σ + θxa q − q dx = a Z ∞−∞ e − β ( x − m ) σ βσ ( σ θa q − β ) + θa q − m q dx = a Z ∞−∞ e βσ ( σ θa q − β ) + θa q − m q e − β ( x − m ) σ e βσ ( σ θa q − β ) + θa q − m q ) q − q dx = ae βσ ( σ θa q − β ) + θa q − m q Z ∞−∞ e − βγ ( x − m ) σ q dx = σ a C q √ β e βσ ( σ θa q − β ) + θa q − m q = e βσ ( σ θa q − β ) + θa q − m q vuut e ( q − βσ ( σ θa q − β ) + θa q − m ) where γ = e ( q βσ ( σ θa q − β ) + θa q − m ) q − and σ = σ γ . . , we have L q ( X )( θ ) = e (cid:16) θma q − − θ a q − σ β (cid:17) − q q = e θm (3 − q )2 a q − − θ a q − σ (3 − q )8 β q
2. We compute the first and the second derivative of L q ( X ) , with respect to θ , weobtain E q ( X ) = L ′ q ( X )(0) E q − ( X ) = 1 q L ” q (0) Intending to interpret these moments, we consider a wonderful property such that anescort distribution obtained by a q-normal distribution with variance σ is equivalent toanother q-normal distribution with q = 2 − q and a variance − qq + 1 σ with ≤ q < . Proposition 5.5.
Let X be a random variable following a q-Gaussian distribution N q ( m, σ ) ,then E q ( X ) = Z + ∞−∞ x [ f ( x )] q ν q ( f ) dx = m V q − ( X ) = σ q − = Z + ∞−∞ ( x − m ) [ f ( x )] q − ν q − ( f ) dx = 3 − qq + 1 σ Proof.
1. Let’s begin with observing a following proportion on a given q-Normaldistribution: q = 2 q − q , σ = β ( q ) β σ = 3 − qq + 1 σ Under this relations, we have √ βσC q (cid:16) e − β ( x − m ) σ q (cid:17) q ν q ∝ e − β ( x − m ) σ qββ q ∝ N q ( m, σ ) Therefore, E q ( X ) = Z + ∞−∞ x [ f ( x )] q ν q ( f ) dx = Z + ∞−∞ xN q ( m, σ )( x ) dx By applying proposition . , we obtain the result.11. The escort function is proportional to the q-gaussian N q ( m, σ ) q = 3 q − q − , σ = β (2 q − β σ = 3 − q q − σ . Hence √ βσC q (cid:16) e − β ( x − m ) σ q (cid:17) q ν q − ∝ e − ( x − m )2 σ q − ββ q ∝ N q ( m, σ ) (6)Then V q − ( X ) = Z + ∞−∞ ( x − m ) [ f ( x )] q − ν q − ( f ) dx = Z + ∞−∞ ( x − m ) N q ( m, σ )( x ) dx = 3 − q − q σ By applying proposition . we obtain the result.In rhis theorem, we prove that for q < a value greater than q +1) (5 q − q − indicates aleptokurtic distribution; a values less than q +1) (5 q − q − indicates a platykurtic distribution.For the sample estimate X , q +1) (5 q − q − is subtracted so that a positive value indicatesleptokurtosis and a negative value indicates platykurtosis. Theorem 5.6.
Let Y be a random variable following a q-Central Gaussian distribution N q (0 , then the coefficient of normalized kurtosis is Kurt [ Y ] = E ( Y )( E ( Y )) = 3( q + 1) (5 q − q − , ≤ q < Proof.
Checking a following proportion on a given q-Normal distribution: q = 5 q − q − , σ = β (4 q − β . ( √ βσC q ) q − e − βx ν q − q ∝ e − x β q − ββ q ∝ N q (0 , σ ) E ( Z ) = Z + ∞−∞ z ( √ βσC q ) q − e − βz ν q − q = Z + ∞−∞ z N q (0 , σ )( z ) dz. Hence, according to Theorem . , we obtain E ( Z ) = 3(3 − q ) (5 − q )(7 − q ) σ ; Z ∼ N q (0 , σ ) . we obtain E ( Z ) = 3(3 − q ) (5 q − q − . Furthermore by observing a following proportion on a given q-Normal distribution: q = 3 q − q − , σ = β (2 q − β ( √ βσC q ) q − e − βx ν q − q ∝ e − x β q − ββ q ∝ N q (0 , σ ) (7)Then E ( Z ) = Z + ∞−∞ z ( √ βσC q ) q − e − βz ν q − q = Z + ∞−∞ z N q (0 , σ )( z ) dz. Hence, according to proposition . , we obtain E ( Z ) = (3 − q ) σ (5 − q ) ; Z ∼ N q (0 , σ ) E ( Z ) = 3 − qq + 1 . q-estimator for random variables are arising from non-extensive statistical mechanics. Inthis section, we will estimate the q-mean and q-variance using the notions of q-Laplacetransform, q-independence.
6. Estimator of q-Mean and q-varianceDefinition 6.1.
Two random variables X and X are said to be q-independent if L q ( X + X )( θ ) = L q ( X )( θ ) ⊗ q L q ( X )( θ ) . efinition 6.2. Let X n be a sequence of identically distributed random variables and m = E ( X ) . Denote S n = n X k =1 X k .. By definition X k , k = 1 , , , ..., is said to be q-independent of the first type (or q-i.i.d.) if for all n = 2 , , , ..., the relations L q [ S n − nm ]( θ ) = L q [ X − m ]( θ ) ⊗ q .... ⊗ q L q [ X n − m ]( θ ) hold. Proposition 6.3.
Let X and X be tow q-independent random variables following re-spectively N q ( m , σ ) and N q ( m , σ ) . Then X + X y N q ( m + m , σ + σ ) Proof. L q ( X + X )( θ ) = L q ( X )( θ ) ⊗ q L q ( X )( θ )= e − q σ + σ ) θ a q − β + − q σ + σ ) θa q − q = (cid:16) e ( σ + σ ) θ a q − β +( σ + σ ) θa q − q (cid:17) − q Note that if X and X are q-Gaussian and q-independent random variables with distri-butions N q ( m , σ ) and N q ( m , σ ) respectively then V ( X + X ) = 3 − q − q ( σ + σ ); 1 q < and V q − ( X + X ) = 3 − qq + 1 ( σ + σ ) = V q − ( X ) + V q − ( X ); In this case cov ( X , X ) = 0 , because V ( X + X ) = V ( X ) + V ( X ) + 2 cov ( X , X ) . Corollary 6.4.
Let X , X , ..., X n be n q-independent random variables with same q-Gaussian distribution N q ( m, σ ) , then X n = n n X i =1 X i follows the q-Gaussian N q ( m, σ n ) . Observe that V ( X ) −→ as n −→ ∞ . Since E ( X ) = m , then the estimates ofm becomes increasingly concentrated around the true population parameter. Such an14stimate is said to be consistent.The empirical variance S n = n P ni =1 ( X i − X n ) is not an unbiased estimate of σ . Indeed E ( S n ) = n − n V ( X )= n − n − q − q σ Therefore b σ = nn − − q − q S n . is an unbiased estimate of σ . Proposition 6.5.
Law of Large Numbers (LLN): If the distribution of the i.i.d. q-independent X , ..., X n is such that X has finite q-expectation, i.e. | E q ( X ) | < ∞ , thenthe sample average X n = X + ... + X n n −→ E q ( X ) converges to its expectation in probability. Theorem 6.6.
Central Limit Theorem (CLT):[18, 10, 19, 4] For q ∈ (1 , , if X , X , ..X n are q-independent and identically distributed with q-mean m q and a finite second (2 q − -moment σ q − , then Z n = X + ... + X n − nm q C q,n,σ q − converges to N q − (0 , Gaussian distribution.
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