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...er, the PGF’s corresponding to this family do not generate commutative semigroup, so that Proposition 1.2 is not applicable in this case. Remark 2. If φ in (1) corresponds to a stable law (see, e.g., =-=[21]-=-), then the distributions corresponding to ψ are called ν-stable laws (see, e.g., [12, 13]). Suppose further that φ corresponds to a strictly stable distribution with index α ∈ (0, 2], so that for all...

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... 2. Then by Proposition 1.1 the ChF of the limiting distribution of the random sums (2) with geometrically distributed νp is ψ(t) = (1+ t 2)−1 - the standard classical Laplace distribution (see, e.g, =-=[11]-=-). Proposition 1.2 now implies that for each p ∈ (0, 1) the ChF of the random sum (8) is (1+ t2/p)−1, which is again a Laplace ChF. We thus obtained the stability property of the Laplace distribution ...

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...ndard exponential as before. Then, in view of Proposition 1.1 we obtain the LT of the limiting distribution of (2) to be ψ(t) = (1+tα)−1. This is the standard MittagLeffler distribution introduced in =-=[19]-=-. The application of Proposition 1.2 shows that we have the stability property with respect to geometric summation as well (as expected, since φ is strictly stable). Example 4. Now take φ(t) = e−|t| α...

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...l variable with the LT λ(t) = (1+ t)−1, and the ChF (or LT for positive variables) of the limiting distribution of the random sums Sp is given by ψ(t) = 1 1− log φ(t) .(4) Since their introduction in =-=[9]-=-, the distributions given by (4) are known as geometrically infinitely divisible laws, as they can be decomposed into geometric convolutions. More precisely, if Y is a RV with the ChF (or LT) (4) abov...

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...istribution with respect to geometric summation (see, e.g., [15]). Let us note that the distributions that arise in these four examples are all special cases of geometric stable (GS) laws (see, e.g., =-=[14]-=-), which are given by the ChF ψ(t) = (1− log φ(t))−1 with a stable ChF φ. These infinitely divisible distributions appear as weak limits of (normalized) sums of independent and identically distributed...

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...f φ(t) = (1+ tα)−1 is the LT of the standard Mittag-Leffler distribution, then by Proposition 1.1 we obtain ψ(t) = (1 + log(1 + tα))−1, which is the geometric Mittag-Leffler distribution discussed in =-=[6]-=-. This recovers Theorem 2.2 of [6]. In turn, Proposition 1.2 shows that for each p ∈ (0, 1) the LT of the random sum (8) is (1+(1/p) log(1+tα))−1, which corresponds to geometric quasi factorial gamma ...

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...)−1. 160 Tomasz J. Kozubowski Example 8. Let φ(t) = (1 + |t|α)−1, 0 < α < 2, be the ChF of the standard Linnik distribution, in which case φp corresponds a generalized Linnik distribution (see, e.g., =-=[3, 17]-=-), also known as generalized α-Laplace variable (see [23]). Again, by Proposition 1.1, we obtain ψ(t) = (1 + log(1 + |t|α))−1, which is the geometric α-Laplace distribution defined in [23]. This recov...

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...andom sum (8) is (1 + t/p)−1, which is again an exponential variable. Thus we recover the well-known stability property of the exponential distribution with respect to geometric summation (see, e.g., =-=[1]-=-). Example 2. Let φ(t) = e−t 2 be the ChF of a normal distribution with mean zero and variance 2. Then by Proposition 1.1 the ChF of the limiting distribution of the random sums (2) with geometrically...

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... via (7), Proposition 1.2 shows that for each p ∈ Δ, the random sum (8) of the Yi’s has the same type of distribution as each of the terms in the sum. Such random stability properties were studied in =-=[2, 8, 22]-=-, among others. Some specific examples will be considered below. Example 1. Consider a trivial case when the distribution of X is concentrated at 1, so that its LT is φ(t) = e−t. Then by Proposition 1...

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...h that we actually have the equality in distribution, Y d = νp∑ j=1 X (p) j .(5) It is worth noting that not all distributions with the ChF (1) admit the random divisibility property (5). As shown in =-=[4, 10]-=-, those that do must have a special structure: the probability generating functions (PGF’s) generated by the family {νp, p ∈ Δ} must form a commutative semigroup with the operation of superposition. I...

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...)−1. 160 Tomasz J. Kozubowski Example 8. Let φ(t) = (1 + |t|α)−1, 0 < α < 2, be the ChF of the standard Linnik distribution, in which case φp corresponds a generalized Linnik distribution (see, e.g., =-=[3, 17]-=-), also known as generalized α-Laplace variable (see [23]). Again, by Proposition 1.1, we obtain ψ(t) = (1 + log(1 + |t|α))−1, which is the geometric α-Laplace distribution defined in [23]. This recov...

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...9 geometric as before. Again, by Proposition 1.1 we obtain ψ(t) = (1 + |t|α)−1, which corresponds to the Linnik distribution (see, e.g., [11] and references therein), also known as α-Laplace law (see =-=[18]-=-). The application of Proposition 1.2 again recovers the well-known stability property of this distribution with respect to geometric summation (see, e.g., [15]). Let us note that the distributions th...

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...hat Proposition 1.2 is not applicable in this case. Remark 2. If φ in (1) corresponds to a stable law (see, e.g., [21]), then the distributions corresponding to ψ are called ν-stable laws (see, e.g., =-=[12, 13]-=-). Suppose further that φ corresponds to a strictly stable distribution with index α ∈ (0, 2], so that for all c > 0 and t ∈ R we have φc(t) = φ(c1/αt). Then, if the PGF’s of the family {νp, p ∈ Δ} an...

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...rein), also known as α-Laplace law (see [18]). The application of Proposition 1.2 again recovers the well-known stability property of this distribution with respect to geometric summation (see, e.g., =-=[15]-=-). Let us note that the distributions that arise in these four examples are all special cases of geometric stable (GS) laws (see, e.g., [14]), which are given by the ChF ψ(t) = (1− log φ(t))−1 with a ...

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...ample 5. If φ(t) = (1+t)−1 is the LT of a standard exponential distribution, then by Proposition 1.1 we obtain ψ(t) = (1 + log(1 + t))−1, which is the geometric exponential distribution introduced in =-=[20]-=- and studied in [7]. We thus recovered Theorem 3.1 of [20] (as well as Theorem 2.1 of [7]). Further, Proposition 1.2 shows that for each p ∈ (0, 1) the LT of the random sum (8) is (1+ (1/p) log(1+ t))...

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...hat Proposition 1.2 is not applicable in this case. Remark 2. If φ in (1) corresponds to a stable law (see, e.g., [21]), then the distributions corresponding to ψ are called ν-stable laws (see, e.g., =-=[12, 13]-=-). Suppose further that φ corresponds to a strictly stable distribution with index α ∈ (0, 2], so that for all c > 0 and t ∈ R we have φc(t) = φ(c1/αt). Then, if the PGF’s of the family {νp, p ∈ Δ} an...

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...n distribution (see [11], Chapter 4) or generalized Laplace distribution (see [16]). Then by Proposition 1.1 we obtain ψ(t) = (1+ log(1+ t2))−1, which is the geometric Laplace distribution defined in =-=[23]-=-. This recovers Theorem 1 of [23]. Similarly, Proposition 1.2 shows that for each p ∈ (0, 1) the ChF of (8) is (1 + (1/p) log(1 + t2))−1. 160 Tomasz J. Kozubowski Example 8. Let φ(t) = (1 + |t|α)−1, 0...

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...he ChF of the standard Laplace distribution, in which case φp corresponds to another ID distribution called Bessel function distribution (see [11], Chapter 4) or generalized Laplace distribution (see =-=[16]-=-). Then by Proposition 1.1 we obtain ψ(t) = (1+ log(1+ t2))−1, which is the geometric Laplace distribution defined in [23]. This recovers Theorem 1 of [23]. Similarly, Proposition 1.2 shows that for e...

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... via (7), Proposition 1.2 shows that for each p ∈ Δ, the random sum (8) of the Yi’s has the same type of distribution as each of the terms in the sum. Such random stability properties were studied in =-=[2, 8, 22]-=-, among others. Some specific examples will be considered below. Example 1. Consider a trivial case when the distribution of X is concentrated at 1, so that its LT is φ(t) = e−t. Then by Proposition 1...

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...1+t)−1 is the LT of a standard exponential distribution, then by Proposition 1.1 we obtain ψ(t) = (1 + log(1 + t))−1, which is the geometric exponential distribution introduced in [20] and studied in =-=[7]-=-. We thus recovered Theorem 3.1 of [20] (as well as Theorem 2.1 of [7]). Further, Proposition 1.2 shows that for each p ∈ (0, 1) the LT of the random sum (8) is (1+ (1/p) log(1+ t))−1, which correspon...