CHARACTERIZATION OF DISTRIBUTIONS WITH THE LENGTH-BIAS SCALING PROPERTY

For q ∈ ( 0,1 ) ﬁxed, we characterize the density functions f of absolutely continuous random variables X > 0 with ﬁnite expectation whose respective distribution functions satisfy the so-called (LBS) length-bias scaling property X L = q b X , where b X is a random variable having the distribution function For an absolutely continuous random variable X > 0 with probability density function (pdf) f and ﬁnite expectation E X , we denote by b X an absolutely continuous random variable having the probability density function ( E X ) − 1 x f ( x ) . In this case, b X is called the size- or length-biased version of X and L ( b X ) is the corresponding length-biased distribution. It is well known that b X is the stationary total lifetime in a renewal process with generic lifetime X (see [ 2, Chapter 5 ] ). The length-biased distributions have been applied in various ﬁelds, such as biometry, ecology, environmental sciences, reliability and survival analysis. A review of these distributions and their applications are included in [ 5, Section 3 ] , [ 6, 8, 12, 13 ] . In [ 9 ] , Pakes and Khattree ask whether it is possible to randomly rescale the total lifetime to recover the lifetime law. More speciﬁcally, let V ≥ 0 be a random variable independent of X with a ﬁxed law satisfying P ( V > 0 ) > 0.

For an absolutely continuous random variable X > 0 with probability density function (pdf) f and finite expectation EX , we denote by X an absolutely continuous random variable having the probability density function (EX ) −1 x f (x) . In this case, X is called the size-or length-biased version of X and ( X ) is the corresponding length-biased distribution. It is well known that X is the stationary total lifetime in a renewal process with generic lifetime X (see [2,Chapter 5]).
The length-biased distributions have been applied in various fields, such as biometry, ecology, environmental sciences, reliability and survival analysis. A review of these distributions and their applications are included in [5,Section 3], [6,8,12,13].
In [9], Pakes and Khattree ask whether it is possible to randomly rescale the total lifetime to recover the lifetime law. More specifically, let V ≥ 0 be a random variable independent of X with a fixed law satisfying P (V > 0) > 0. For which laws (X ) does the following "equality in law" hold? For instance, when V has the uniform law on [0, 1] the last equality holds if and only if (X ) is an exponential law (see [9]).
In this note we consider the case where V is a constant function: The law of X has the so-called length-bias scaling property (abbreviated to LBS-property) if 186 DOI: 10.1214/ECP.v14-1458 with q ∈ (0, 1). Several authors, including Chihara [3], Pakes and Khattree [9], Pakes [10,11], Vardi et al. [14], have studied the LBS-property. In [1], Bertoin et al. analyze a random variable X that arises in the study of exponential functionals of Poisson processes; they show that q X = X = q −1 X −1 , with EX = q −1 .
An easy computation shows that (1) can be written as By induction we have that When X is an absolutely continuous random variable with probability density function f , we sometimes write X ∼ f .

Proposition 1.
If X ∼ f and f satisfies (2), then the pdf g (x) = e x f (e x ) of the random variable Y = log X satisfies the functional equation So, the main result of this note characterizes the probability density functions fulfilling the last functional equation. First, we recall that the theta function given by for all (x, t) ∈ 2 + , satisfies the heat equation on 2 + and Theorem 1. Let a, b, C be real numbers with a b > 0, C > 0. Then the pdf g satisfies the functional equation (4) if and only if there exists a 1-periodic function ϕ, ϕ ≥ −1, such that the restriction of ϕ to (0, 1) belongs to L 1 (0, 1), where −µ = ln C + b/2, satisfies the functional equation (4) with a = 1. If the density function g so does, then g By using (6), the result follows with ϕ ( where µ = ln q 1/2 EX . In [10, Theorem 3.1], Pakes uses a different approach to characterize the probability distribution functions F = (X ) satisfying (1) with EX = 1.
By (3), it follows that the probability density functions having the LBS-property are solutions of an indeterminate moment problem.
Let N µ, − ln q be the normal density with mean µ and variance − ln q. If Y ∼ N µ, − ln q , we note that exp(Y ) has the log-normal density, i.e. exp(Y ) ∼ 1 x −2π ln q exp ln x − µ 2 2 ln q .

Remark 1. If X is a positive absolutely continuous random variable with pdf f , then
So, for ν ∈ the distributional identity X = e 2ν X −1 is equivalent to the functional equation If ϕ is a measurable function on and f is a pdf function given as follows x > 0, then f satisfies the latter functional equation if and only if ϕ is an even function.
As a consequence of Corollary 1 and the last remark with ν = ln q 1/2 EX , we have that a positive random variable X with probability density function f satisfies if and only if f can be written as in Corollary 1 with ϕ being an even function.
Finally, we provide some families of functions satisfying (8).

Examples
From bounded functions, the following observation allows to construct functions with values in the non-negative axis.
if and only if ϕ (x) is orthogonal to θ (x, t) in L 2 (0, 1). By (5) this is equivalent to the orthogonality between c n n∈ and e −4π 2 n 2 t n∈ , i.e. n∈ c n e −4π 2 n 2 t = 0.
In [11, page 1278] Pakes says that the continuous solutions of (2) probably are exceptions. But for any trigonometric polynomial p (x) = |n|≤N c n e 2πni x whose coefficients c n ∈ satisfy the last equality with t = b −1 /2, there is an interval I such that ǫ x ≥ −1, for all x ∈ min Re p, max Re p , ǫ ∈ I, therefore ϕ = ǫRe p ≥ −1 on and the corresponding density function given by Corollary 1 is an infinitely differentiable function on + . Example 4. By (6) we have that is a 1-periodic, continuous function satisfying (11) for all c ∈ [0, 1). Since θ (x, t) is an even function for all t > 0, the function ϕ c is even if and only if c = 0, 1/2. In [4, equality (2.15)] it is shown that where q t = e −t −1 /2 , p; q ∞ = To get more examples of functions fulfilling (8) see [4]. The results in [7] can be used to construct positive random variables having not the LBS-property but with moment sequence (3).