Generalized convolutions and the Levi-Civita functional equation

In Borowiecka et al. (Bernoulli 21(4):2513–2551, 2015) the authors show that every generalized convolution can be used to define a Markov process, which can be treated as a Lévy process in the sense of this convolution. The Bessel process is the best known example here. In this paper we present new classes of regular generalized convolutions enlarging the class of such Markov processes. We give here a full characterization of such generalized convolutions $$\diamond $$⋄ for which $$\delta _x \diamond \delta _1$$δx⋄δ1, $$x \in [0,1]$$x∈[0,1], is a convex linear combination of $$n=3$$n=3 fixed measures and only the coefficients of the linear combination depend on x. For $$n=2$$n=2 it was shown in Jasiulis-Goldyn and Misiewicz (J Theor Probab 24(3):746–755, 2011) that such a convolution is unique (up to the scale and power parameters). We show also that characterizing such convolutions for $$n \geqslant 3$$n⩾3 is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions.


Motivations
Generalized convolutions were invented and studied by K. Urbanik (see [11][12][13][14][15]). The idea was taken from the paper of Kingman [5], who introduced and studied special cases of such convolutions now called Kingman convolutions or Bessel convolutions. In the simplest case Kingman's work was based on an obvious observation that rotationally invariant distributions in R n form a convex weakly closed set with the extreme points {T a ω n : a 0}, where ω n is the uniform distribution on the unit sphere S n−1 ⊂ R n , T a is the rescaling operator, i.e. T a λ is the distribution of aX if λ is the distribution of X (abbreviation: λ = L(X)).
Kingman was working on one-dimensional projections of ω n and he found all distributions λ, λ = L(θ) for which aθ + bθ d = a 2 + b 2 + 2abR θ, a, b > 0, for some fixed, but dependent on n, random variable R independent of θ. Here d = denotes equality of distributions and θ is an independent copy of θ. In this case the generalized convolution is defined by the formula Urbanik noticed that the Kingman convolution is a special case of generalized convolutions, i.e. associative, symmetric, weakly continuous linear operators : P 2 + → P + (here P + denotes the set of all probability measures on [0, ∞)) for which λ δ 0 = λ, λ (pλ 1 + (1 − p)λ 2 ) = pλ λ 1 + (1 − p)λ λ 2 , T a (λ 1 λ 2 ) = (T a λ 1 ) (T a λ 2 ). For some technical reasons Urbanik assumed also that there exists a sequence of positive numbers (a n ) such that T an δ n 1 converges weakly to some non-degenerate to δ 0 measure. This assumption is not necessary in most of the results.
We see that generalized convolutions extend in, the language of distributions, the idea of sums of independent random variables. It was shown in [1] that if we restrict our attention to generalized sums of independent random variables considering ⊕ as an associative, symmetric operation for which a(X ⊕ Y ) = (aX) ⊕ (aY ), then we have only two possibilities: • X ⊕ Y = X α + Y α 1/α in the case of positive variables, • X ⊕ Y = X <α> + Y <α> <1/α> for variables taking values in R.
Here α can be any number from the set (0, ∞] and x <α> := |x| α sign(x). Even the Kingman convolution cannot be written in this way as it requires the assistance of an extra variable R. Considering generalized convolutions instead of generalized sums enrich the theory significantly.
The introduction of generalized convolutions required very laborious and time consuming introductory studies before the theory was read to define stochastic processes in the sense of generalized convolutions and before they could be used in stochastic modeling and other applications. This was done in a series of papers by many authors, see e.g. [9,[11][12][13][14][15] In the paper [1] the authors defined, proved the existence of and studied properties of stochastic processes with independent increments in the sense of generalized convolutions and the corresponding stochastic integrals. Some of these constructions were given earlier by Thu [9,10] in a special case of Bessel convolutions.
In this paper we focus on constructing new examples of generalized convolutions with the special property

Preliminaries
According to the Urbanik paper (see [11]) a commutative and associative P +valued binary operation defined on P 2 + is called a generalized convolution if for all λ, λ 1 , λ 2 ∈ P + and a 0 we have: (iv) if λ n → λ, then λ n η → λ η for all η ∈ P and λ n ∈ P + , (v) there exists a sequence (c n ) n∈N of positive numbers such that the sequence T cn δ n 1 converges to a measure different from δ 0 ; where → denotes the weak convergence of probability measures.
A pair (P + , ) is called a generalized convolution algebra. It has been proven in [15] (Theorem 4.1 and Corollary 4.4) that each generalized convolution admits a weak characteristic function, i.e. a one-to-one correspondence λ ↔ Φ λ between measures λ from P + and real-valued Borel functions Φ λ from L ∞ (m 0 ), m 0 = δ 0 + , where is the Lebesgue measure on (0, ∞), so that The function ϕ : R + → R, defined by ϕ(t) = Φ δt (1) = Φ δ1 (t) is called a probability kernel of (P + , ). The kernel ϕ is a Borel function, ϕ(0) = 1 and |ϕ(t)| 1 for each t ∈ [0, ∞). It is evident that A generalized convolution algebra (P + , ) (and the corresponding generalized convolution ) is said to be regular if its probability kernel ϕ is a continuous function. It is known by [11], p.219, that the max-convolution introduced by the operation max(X, Y ) on independent random variables X and Y is not regular and its probability kernel is given by The -generalized characteristic function in a generalized convolution algebra plays the same role as the classical Laplace or Fourier transform for convolutions defined by addition of independent random elements.
The following proposition shows how we can get a new generalized convolution using an already known one. This result is not especially deep however it will be useful in further considerations.

Proposition 1.
Assume that a non-trivial generalized convolution algebra (P + , ) admits a characteristic function Φ with the probability kernel ϕ. Then for every α > 0 there exists a generalized convolution on P + with the generalized characteristic function where L(Y ) denotes the distribution of the random variable Y .
Proof. It is enough to define the generalized convolution on the measures δ x , δ y for x, y 0. Assume that δ x α δ y α = L(Z) for some nonnegative random variable Z. We see that Now we are able to define the generalized convolution : Checking that is a generalized convolution and that Ψ is the generalized characteristic function for the algebra (P + , ) is trivial and will be omitted.

Main problem
We want to characterize such general convolutions for which the convolution of two one-point measures δ x , δ 1 is a convex linear combination of n fixed measures and only the coefficients of this linear combination depend on x.
Let ϕ be the kernel (unknown) of the considered generalized convolution and In the language of generalized characteristic functions our problem leads to the following functional equation Remark 2. Without loss of generality we can assume that λ 1 ({1}) = · · · = λ n−1 ({1}) = 0. If this is not the case then we put for some q k ∈ [0, 1], k = 1, . . . n, such that λ k ({1}) = 0 and then we can write for q 0 = 1 Remark 3. Notice that if the measures λ 0 , . . . , λ n−1 are linearly dependent, i.e. one or more measures can be obtained as a convex linear combination of others then equality ( * ) can be written as for some m < n−1 and some probability measures λ 1 , . . . , λ m . From now on we will assume that λ 0 , . . . , λ n−1 are linearly independent. It means also that their generalized characteristic functions ϕ, Φ 1 , . . . , Φ n−1 are linearly independent.
We will show that under some additional assumptions equation ( * * ) can be written in the form of the multiplicative Levi-Civita functional equation, which is described in the following theorem (for details see e.g. [8]).

Theorem 1. Let a complex-valued continuous function ϕ satisfy the equation
for all x, y ∈ (0, 1) where P j are polynomials and λ j ∈ C. Proof. Let t > 0. If there exists a sequence (a n ), a n → ∞ for n → ∞, such that lim n→∞ ϕ(ta n ) = c = 0 then we have We can choose n 0 large enough to have |ϕ(ta n )| > |c|/2 for n n 0 . Then |Φ k (ta n )/ϕ(ta n )| < 2/|c|. Since Proof. Of course a > 0 since ϕ(0) = 1 and ϕ is a continuous function. Assume that a = ∞. We have that ϕ(t) > 0, ϕ(xt) > 0 for every t > 0 and x ∈ [0, 1]. In equation ( * * ) we can divide both sides by ϕ(t) and obtain If we restrict the argument t to the interval (0, 1) we get a version of the Levi-Civita functional equation We can apply now Theorem 1 and obtain that for some M ∈ N, λ j ∈ C, and some polynomials P j . Since our function ϕ is real as a generalized characteristic function, we have that λ 1 , . . . , λ M are real.
Considering the function ϕ c (·) := ϕ(c·) for c > 0 we see that thus, using Theorem 1 again, we obtain that for some M c ∈ N, λ j, c ∈ R, and some polynomials P j, c . Consequently The functions ϕ| (0,1] and ϕ| (0,c −1 ] coincide on the interval (0, 1] for c < 1, thus for every c < 1 Letting c 0 we obtain that for some M ∈ N, λ j ∈ R, and some polynomials P j , j ∈ {1, . . . , M} In order to discuss the limit behavior of the function ϕ around zero and infinity we substitute t → e −x and obtain Let r = max{λ j : j M } and s = min{λ j : j M }. It is easy to see now that if r > 0 then |ϕ(e −x )| → ∞ if x → ∞, which is impossible since any generalized characteristic function is bounded. If s < 0 then |ϕ(e −x )| → ∞ if x → −∞, which is impossible for the same reason. Thus we have that r = s = 0 and ϕ is a polynomial bounded on (0, ∞). This however is possible only if ϕ is a constant function in contradiction to our assumptions.
Example 1. The convolutions described in this example were introduced by J. Kucharczak and K. Urbanik in [6]. If ϕ n (t) = (1 − t α ) n + then for all x ∈ [0, 1] and t 0 We see that ϕ n is a solution of the Levi-Civita equation ( * * * ), but in order to see that it is also a solution of equation ( * * ) we need to show that for each k = 1, . . . , n there exists a measure λ k,n with distribution function F k,n such that It is easy to see that for λ 1,n = π α(n+1) , where π c is the Pareto distribution with density function g c (s) = cs −c−1 1 [1,∞) (s), we have We see now that λ k,n = L(Z 1,n . . . Z n,n ) where Z 1,n . . . Z n,n are independent and L(Z k,n ) = π α(n+k) . It is only a matter of laborious calculations to show that λ k,n , k 1 has density function Of course λ 0,n = δ 1 . Consequently ϕ n is a solution of equation ( * * ). The formal definition of this convolution for x ∈ [0, 1] can be written in the following form: Example 2. In [16] K. Urbanik gave an example of a not regular generalized convolution different from the max-convolution. It is called (1, p)-convolution with p ∈ (0, 1) and defined for p = 1 2 by Notice that we have here a solution of equation ( * * ) with n = 2, p 0 (x) = (1 − px), p 1 (x) = px and the probability kernel given by Notice that ϕ here is not continuous at 1 as a function on the whole [0, ∞) and discontinuity appears only at this point.

Applying the solution of the Levi-Civita equation for n = 3
The main aim of this paper is to show that for n > 2 there exist more than one solution of equation ( * * ) in the set of generalized characteristic functions. We show this in the case n = 3 under the following additional assumptions: The assumption p 1 (1) = 1 implies that Φ 1 (t) = ϕ(t) 2 . Since ϕ(t) = 0 for each t ∈ [0, 1) equation ( * * ) can be written in the following way: By the continuity of generalized characteristic functions we see that thus the limit g = lim t→1 − Φ2(t) ϕ(t) exists anyway, but the additional assumption g = 0 is equivalent to the condition p 0 (x) = ϕ(x). Consequently, equation ( * * ) restricted to the set [0, 1] under the additional assumptions (A) can be written as the following version of the Levi-Civita functional equation:
By Proposition 1, without loss of generality, we can assume that α = 1 and p > 1, thus we shall discuss only the following functions: It turns out that only one type of such functions is admissible for us. Proof. We show that there is no cumulative distribution function F for which implies that F (s) = 0 for all s < 1. We see now that the function F satisfies the following equation: where by the Stieltjes integral b a g dF we understand [a,b) g dF . Notice now that for a continuous differentiable function g and a cumulative distribution function F such that F (1) = 0 < F(1 + ) we have the following formula for integration by parts for every continuity (with respect to Using this formula, dividing both sides of our equation by t and substituting t −1 = u we get for almost every u > 1 The left hand side of this equation is differentiable for each u > 1 thus also the right hand side is differentiable for u > 1 and we get Case 1. If c = 1 then we obtain We see that F (1 + ) = 0, lim s→∞ F (u) = 1, as it shall be expected, but the corresponding density function can take negative values: Since F 1 (1 + ) = 0, K = 1 and the eventual density function f = F would be the following: This however is impossible since the expression in brackets is negative for u large enough. Case 3. If c ∈ {1, 1 2 } then for some K we have By Remark 2 we can assume that F 1 (1 + ) = 0, thus K = 2c(1−c) 3 (2c−1) 3 . Now we see that the eventual density function would be the following: If c > 1 2 then 1−β < 0 thus the expression in the brackets is close to − c 2 2c−1 < 0 for u large enough, thus f cannot be a density function for any probability distribution. If c ∈ (0, 1 2 ) then 1 − β > 0 and K < 0, thus the expression in the brackets has the same limit at infinity as which is also impossible for any probability density function.
Considering the probability kernel ϕ(t) = (1 − (c + 1)t + ct p ) + we will show that for n = 3 there exist generalized convolutions defined by equation ( * ) other than the Kucharczak-Urbanik convolutions described in Example 1.
Proof. Let x ∈ [0, 1]. We need to calculate the function F x for which the following equality holds: The function L is zero if xt > 1 or t > 1, thus the integral R vanishes if 1/t < 1. This means that the distribution function F x is supported on [1, ∞).
For t < 1 integrating by parts we obtain Substituting t = u −1 > 1 we have that Applying the same operations to the function L we have where λ 1 , λ 2 are probability measures with densities If c = (p − 1) −1 and p ∈ (1, 2) then none of the functions ϕ c,p can be a probability kernel of any generalized characteristic function.
Proof. Applying Lemma 3 we see that for c = (p − 1) −1 we have 1 + c − pc = 0. Comparing u p (uR) = u p (uL) we obtain Consequently for u 1 The function F x is a cumulative distribution function of some measure Λ x . We see that F x (+∞) = 1 thus Λ x ([1, ∞)) = 1 and which means that the measure Λ x has an atom at the point 1 of the weight ϕ(x). Moreover, Case 1. If p 2 it is enough to notice that x > x p , u −p−1 > u −2p and u −2 > u −2p , and we obtain x p u 2p = 0, which shows that λ x is a positive measure. In order to get the final formulation of Proposition 4 it is enough to notice that for u 1 where is the cumulative distribution function of the measure is the cumulative distribution function for the measure λ 1 (du) = 2p u −p−3 (p − 1) 2 (p + 1)u + (p − 2)u p − (2p − 1)u 2−p 1 [1,∞) (u) du.
We need to check that λ 1 is a positive measure. To see this it is enough to notice that in the last formula the expression in the brackets for u > 0 is greater than (p + 1) + (p − 2) − (p − 1) = p > 0. It is evident that F 1 (+∞) = F 2 (+∞) = 1 thus λ 1 , λ 2 are probability measures, which ends the proof in the case p 2. Case 2. If p ∈ (1, 2) we can write This means that lim u→∞ u 3 F x (u) = 2p(p−2)x (p−1) 2 < 0 thus F x (u) is negative at least for u large enough and x = 0, and it cannot be a density function for any positive measure.