The chi-square distribution with n degrees of freedom has an important role in probability, statistics and various applied fields as a special probability distribution. This paper concerns the relations between geometric random sums and chi-square type distributions whose degrees of freedom are geometric random variables.
Science & Technology Development Journal, 22(1):180- 184 Research Article On chi-square type distributions with geometric degrees of freedom in relation to geometric sums Tran Loc Hung∗ ABSTRACT The chi-square distribution with n degrees of freedom has an important role in probability, statistics and various applied fields as a special probability distribution This paper concerns the relations between geometric random sums and chi-square type distributions whose degrees of freedom are geometric random variables Some characterizations of chi-square type random variables with geometric degrees of freedom are calculated Moreover, several weak limit theorems for the sequences of chi-square type random variables with geometric random degrees of freedom are established via asymptotic behaviors of normalized geometric random sums MSC2010: Primary 60E05; 60E07; Secondary 60F05; 60G50 Key words: Chi-square distribution, Geometric random sums, Weak limit theorems INTRODUCTION Let {Xn , n ≥ 1} be a sequence of independent, standard normal distributed random variables, (shortly, Xn ∼ N (0, 1), n ≥ 1) It has long been known that the partial sum X12 +X22 + · · · + Xn2 is said to be a chi-square random variable with n degrees of freedom, denoted by χ (n) The probability density function of the χ (n) is given by University of Finance and Marketing Correspondence Tran Loc Hung, University of Finance and Marketing Email: tlhung@ufm.edu.vn History • Received: 2018-11-21 • Accepted: 2019-03-24 • Published: 2019-03-31 DOI : https://doi.org/10.32508/stdj.v22i1.1053 Copyright © VNU-HCM Press This is an openaccess article distributed under the terms of the Creative Commons Attribution 4.0 International license f (x) = χ (n) e−x/2 xn/2−1 , 2n/2 Γ(n/2) 0, for x > 0, (1) for x ≤ 0, ∫ where Γ(y) = 0+∞ e−x xy−1 dx (for y > 1), denotes the Gamma function (see { } for instance ) It is eas2 ily seen that X j , j ≥ is a sequence of independent, identically distributed (i.i.d.) random ( ) variables, X j2 ∼ χ (1) for j ≥ with mean E X j2 = and fi( ) nite variance Var X j2 = 2, for all j ≥ Thus, the chi-square random variable χ (n) should be considered as a partial sum of n desired i.i.d random variables X j2 , j ≥ Especially, degree of freedom of chi-square distribution χ (n) is a deterministic number n of square of i.i.d random variables having standard normal distribution in summation It is also worth{pointing out } that for large n the desired sequence X j2 , j ≥ will be obeyed the classical weak limit theorems like weak law of large numbers and central limit theorem Especially, the classi- cal Weak law of large numbers states that ( ) χ (n) − E χ (n) = n P n −1 n ∑ j−1 X j → as n → ∞ (2) or n−1 ∑nj=1 X j2 → D1 as n → ∞, D (3) where D1 is a random variable degenerated at point Furthermore, the Central limit theorem will be formulated as follow ( ) χ (n) − E χ (n) )]1/2 = [ ( Var χ (n) ( ) (4) Xj − D n −1/2 √ n → N (0, 1)asn → ∞ ∑ j=1 (see for instance , page 156-159) Here and subP D sequently, the symbols and stand for the → → convergence in probability and convergence in distribution, respectively The chi-square distribution with degrees of freedom n plays an important role in various applied problems like χ − testing in nonparametric statistics, in estimation theory or in testing hypothesis, etc (see for more details) The interesting question arises as to what happens with the distribution of chi-square random variable with n degrees of freedom, when the deterministic number n (degree of freedom) will be replaced by a positive-integer valued random variable N, which independent of all Cite this article : Loc Hung T On chi-square type distributions with geometric degrees of freedom in relation to geometric sums Sci Tech Dev J.; 22(1):180-184 180 Science & Technology Development Journal, 22(1):180-184 Xn , n ≥ This question has been addressed in the article Moreover, the results should be more interesting if the degree of freedom being a geometric random variable N p , p ∈ (0, 1), independent of all X j , j ≥ and having a probability mass function ( ) P N p = k = p(1 − p)k−1 , k ≥ 1, p ∈ (0, 1) Then the sum X12 + X22 + · · · + XN2 p of random variables Xn2 up to the geometric degrees of freedom, denoted by ( ) ( ) χ N p On the other hand, the χ N p may be considered in the role of a compound geometric ( ) sum χ N p := X12 + X22 + · · · + XN2 p , which will lead to interesting results, too Itshould be noted that in the classical literature the compound geometric sums have been attracting much attention Actually, the compound geometric sums can model many phenomena in insurance, queuing, finances, reliability, biology, storage, and other real world fields (for a deeper discussion of this the reader is referred to ) This paper deals with study of the distribution of chisquare type random variables with geometric degrees of freedom via geometric random sums Some characterizations of the ( ) χ N p are given Two asymptotic results of the ( ) probability distribution functions of the χ N p are also investigated in two limit theorems for compound geometric sums The organization of this paper is as follow Section deals with some ( ) characterizations of the χ N p An algorithm of cal( ) culating the probability density function of χ N p is presented in this section In Section the asymptotic behavior of desired normalized sum the asymptotic behaviors of two normalized random ] [ 2geometric ( ) X j −1 Np 1/2 when p ↘ 0+ sums pχ N p and p ∑ j=1 √ Proposition 2.1 The density probability function ( ) of χ N p is given by ∞ ( ) f χ (Np ) (x) = ∑ P N p = n f χ (n) (x) n=1 = ∞ ∑ p(1 − p)n−1 f χ (n) (x), (5) n=1 x ∈ (0, +∞) According to the formula in (5), the probability den2 sity function of the χN(p) should be calculated by following algorithm Algorithm 2.1 Define the probability tion f χ (n) (x)in (1) distribution func- ( ) Compute the probabilitiesP N p = n = p(1 − p)n−1 , n ≥ 1related to the geometric random variableN p with parameter p ∈ (0, 1) Compute the probability distribution function f χ (Np ) (x)with the geometric degrees of freedomN p , by the formula (5) ∞ f χ (Np ) (x) = ∑ p(1 − p)n−1 f χ (n) (x) n=1 Proposition 2.2 The probability distribution func( ) tion of χ N p is defined as follows ( ) ( ) Fχ (Np )(x) = ∑∞ n=1 P N p = n P χ (n) ≤ x n−1 F = ∑∞ χ (n) (x), n=1 p(1 − p) x ∈ (0, +∞), ∫ whereF χ (n) (x) = 0x f χ (n) (x)dx (6) CHARACTERIZATIONS OF CHI-SQUARED TYPE RANDOM VARIABLE WITH GEOMETRIC DEGREES OF FREEDOM χ (N p ) According to X j ∼ N(0,1) , for j ≥ 1, hence X j2 ∼ χ (1) for j ≥ Then the numeric characterizations of chi-square type random variable with geometric ( ) degree of freedom χ N p should be directly calculated as follows: Proposition 2.3 Using the Wild’s identity for a random sum (see for instance 10 , the mean of ( ) χ N p should be given in from ( ) ( ( )) ( ) (7) E χ N p = E N p × E X j2 = p−1 For the sake of convenience, we denote by f χ (Np ) (x) and Fχ (Np ) (x) the probability density function and probability distribution of the chi-square type with geometric random ( ) degree of freedom χ N p , respectively Based on formula in (1), the following propositions will be stated without proofs as follows: ( ) The variance of χ N p will be computed by ( ) ( ) ( ( )) Var χ N p = E N p × Var X j2 ( ( ))2 ( ) + E X j2 × Var N p ( ) 1− p 1+ p = 2p−1 + = p2 p will be presented in two weak limit theorems for compound geometric sums of squares of independent standard normal random variables The received results in this paper are a continuation of the 181 (8) Science & Technology Development Journal, 22(1):180-184 Figure 1: Plot of probability density function f χ (Np ) (x) corresponding the geometric parameters p, p ∈ (0, 1), established by formula (5) The following figure is showing the behaviors of curves of the probability density functions defined in (5), corresponding various value of parameter p ∈ (0, 1) Remark 2.1 It is clear that, according to the Figure 1, the curves of the probability density distribution f χ (Np ) (x)are decreasing when values of the parameters p tend to zero This does not allow us to have analogues as asymptotic behaviors of the probability density distribution f χ (n) (x)of the chi-square random variable with geometric degrees of freedomχ (n)in (1) when n tends to infinity (see for more details) The essence of this difference will be explained by weak limit theorems for geometric random sums in next section ASYMPTOTIC BEHAVIORS OF χ (N p ) IN RELATION TO GEOMETRIC RANDOM SUMS Here and subsequently, denote by Em the exponential distributed random variable with mean E (Em ) = , and m, with characteristic function φεm (t) = 1−it D(a) stands for the random variable degenerated ( ) at point a ∈ (−∞, +∞), i.e P D(a) = a = ( ) and P D(a) ̸= a = The following theorems will demonstrate the asymptotic behaviors of two ( ) normalized[ geometric random sums pχ N p and ] N p p1/2 ∑ j=1 X j2 −1 √ when p ↘ 0+ The received results will show the difference between of limiting distributions of normalized geometric random sums and determined sums in terms of assertions (3) and (4) Before stating the main results of this section we first provide some propositions as follows Proposition 3.1 LetEm be an exponential distributed random variable with mean m Then, D N ( j) p εm = p ∑ j=1 εm , (9) ( j) whereEm are i.i.d random variables having exponential distribution with mean m, and independent of N p for p ∈ (0, 1) Here and from now on the notaD tion=stands the identity in distribution Proof According to Theorems 9.1 and 9.2 in 10 (page Np ( j) εm 193-194), the characteristic function of p ∑ j=1 will be defined as follows ( ) φ pΣNp ε ( j) (t) = hNp φε ( j) (pt) = m j=1 m ( j) pφεm (pt) p = − (1 − p)φε ( j) (pt) φ −1 (pt) − + p (10) εm−1 m pm = = m − it − m + pm m = φεm (t) for t ∈ (−∞, +∞), m − it ( ) where hNp (t) = E t Np denotes the probability generating function of N p The Eq (10) finishes the proof 182 Science & Technology Development Journal, 22(1):180-184 Theorem 3.1 Let {Xn n ≥ 1} be a sequence of independent, standard normal distributed random variables Xn ∼ N(0,1) for n ≥ Let N p be a geometric distributed random variable with parameter p, p ∈ (0, 1) Assume that the random variables X1 , X2 , and N p are independent Then, ( ) Np D pχ N p = p ∑ j=1 X j2 → ε1 as p ↘ 0+ , (11) where ε1 ∼ Exp(1) is an exponential distributed random variable with mean and P (ε1 ≤ x) = − e−x for x ≥ Proof Let us denote by hNp (t) := ( 2) ( ) E t Np and φXn2 (t) := E eitXn the probability generating function of N p and the characteristic function of a random variable Xn , respectively Then, direct computation shows that pt , hNp (t) = − (1 − p)t for |t| < and 1−p , p ∈ (0, 1) , φXn2 (t) = (1 − 2it)−1/2 for −∞ < t < +∞, n ≥ In view of theorems 9.1 and 9.2 in 10 (page 193-194), ( ) the characteristic function of the pχ N p is given by Proposition 3.2 The Laplace distributed random variableL(0,1) with zero location parameter and unit scale parameter should be presented in following form D N ( j) p L(0,1) = p1/2 ∑ j=1 L(0,1) , (12) ( j) whereL(0,1) , j ≥ 1are i.i.d Laplace distributed random variables with parameters and 1, independent of N p for p ∈ (0, 1) Proof We shall begin with showing that the charac( j) teristic function of L(0,1) at point p1/2 t is given by ( j) φL (0,1) ( )−1 ) ( 1 p t = + pt 2 Then ( ( )) (t) f = h p2t = ( j) Np Np ( j) L(0,1) ∑ j=1 L(0,1) ( ) ( )−1 pφL(0,1) p t ( ) = + t2 ( j) − (1 − p) fL p2t φ p2 (0,1) = φL(0,1) (t) for t ∈ (−∞, +∞) According to the continuity theorem for characteristic function ( 10 , Theorem 9.1, page 238), the proof is ) ( pφXn2 (pt) φ pχ (Np ) (t) = hNp φXn2 (pt) = = finished − (1 − p)φXn2 (pt) Theorem 3.2 Let the assumptions of the Theorem 3.1 √ p p[ − 2ipt + (1 − p)] √ = = hold Then − 2ipt − (1 − p)2 − 2ipt − (1 − p) √ [ ] − 2ipt + − p Np X − 1 D j √ p ∑ → L(0,1) as p ↘ 0+ − 2it − p j=1 Letting p → 0+ , we can assert that φ pχ (Np ) (t) → (1 − it)−1 = φε1 (t) for all t ∈ (−∞, +∞) whereL(0,1) stands for the Laplace distributed random variable with parameters and 1, having characteristic ) ( In view of the continuity theorem for characteristic function in formφL (t) = + t −1 (0,1) function (see 10 for more details), the proof is finished Proof Without loss of generality we may assume that Remarks 3.1 Theorem 3.1 is an analog of the Rényi’s result (1957) on asymptotic behavior of geometric ranx2j − dom sum of independent, identically positive-valued √ = W j2 for j ≥ random variables with positive mean (see and for more details) ( ) ( ) It makes sense to consider that the assertion in (4) Then, for j ≥ 1, we have E W = and D W = j j will not be valid if the non-random number n (being Using Maclaurin series for characteristic function ( ) degrees of freedom) is replaced by a geometric ran- φ p 12 t , we have Wj dom variable N p , p ∈ (0, 1) The next thereom 3.2 will present the asymptotic[behavior ] of a normalized geo( 2) ( ) pt X −1 N p j metric sum p1/2 ∑ j=1 √ , when p ↘ 0+ φW p t = − pt + o j 2 183 Science & Technology Development Journal, 22(1):180-184 It can be verified that COMPETING INTERESTS Np (t) = E eit p ∑ j=1 W j = None of the authors reported any conflict interest related to this study φ Np p ∑ j=1 W j2 REFERENCES pφW p t j 1 − (1 − p)φW p t (13) j = p = −1 φW p t −1+ p j p [ ( )]−1 pt − pt + o −1+ p 2 Letting p ↘ 0+ , from (13), it follows that ( )−1 φ Np (t) → + t for t ∈ (−∞, +∞) p ∑ j=1 W j In view of the continuity theorem for characteristic function (see 10 for more details), the proof is complete Hogg RV, McKean J, Craig AT Introduction to Mathematical Statistics Seventh Edition, Pearson; 2013 Hung TL, Thanh TT, Vu BQ Some results related to distribution functions of chi-square type with random degrees of freedom Bulletin of the Korean Mathematical Society 2008;45(3):509– 522 Asmusen S Applied Probability and Queues Springer; 2003 Asmusen S Riun Probabilities World Scientific; 2010 Kruglov VM, Korolev VY Limit Theorems for Random Sums, Moskov.Gos Univ., Moscow,; 1990 Kalashnikov V Geometric sums: bounds for rare events with applications Risk analysis, reliability, queuing Mathematics and its Applications, 413 Dordrecht: Kluwer Academic Publishers Group; 1997 Gnedenko BV, Korolev VY Random Summations: Limit Theorems and Applications New York: CRC Press; 1996 Bon JL Geometric Sums in Reliability Evaluation of Regenerative Systems Information Processes 2002;2(2):161–163 Grandell J Risk Theory and Geometric Sums Information Processes 2002;2(2):180–181 10 Gut A Probability: a graduate course Springer Texts in Statistics New York: Springer; 2005 184 ... AT Introduction to Mathematical Statistics Seventh Edition, Pearson; 2013 Hung TL, Thanh TT, Vu BQ Some results related to distribution functions of chi-square type with random degrees of freedom. .. probability distribution of the chi-square type with geometric random ( ) degree of freedom χ N p , respectively Based on formula in (1), the following propositions will be stated without proofs as follows:... CHARACTERIZATIONS OF CHI-SQUARED TYPE RANDOM VARIABLE WITH GEOMETRIC DEGREES OF FREEDOM χ (N p ) According to X j ∼ N(0,1) , for j ≥ 1, hence X j2 ∼ χ (1) for j ≥ Then the numeric characterizations of chi-square