Classes of measures closed under mixing and convolution. Weak stability

Misiewicz, Jolanta K.; Oleszkiewicz, Krzysztof; Urbanik, K.

doi:10.4064/sm167-3-1

Cited by 22 publications

(60 citation statements)

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“…Moreover we know that (Th. 6 in [11]) if µ is weakly stable probability measure on a separable Banach space E then either there exists a ∈ E such that µ = δ a , or there exists a ∈ E \ {0} such that µ = 1 2 (δ a + δ −a ), or µ({a}) = 0 for every a ∈ E. Many interesting classes of weakly stable distributions are already known in the literature. Symmetric stable random vectors are weakly stable, strictly stable vectors are weakly stable on [0, ∞).…”

Section: J K Misiewiczmentioning

confidence: 99%

“…The most important Theorem 1 in the paper [11] states that the distribution µ is a weakly stable if and only if for every a, b ∈ R there exists a probability distribution λ ∈ P such that T a µ * T b µ = µ • λ. Moreover we know that (Th.…”

Section: J K Misiewiczmentioning

confidence: 99%

“…[9,16,17,18]). Recently there appeared a paper written by Misiewicz, Oleszkiewicz and Urbanik (see [11]), where one can find a full characterization of weakly stable distributions with non-trivial discrete part, and a substantial attempt to characterize weakly stable distributions in general case.…”

Section: Introductionmentioning

confidence: 99%

“…This is equivalent (see [11]) with the condition that for all random variables Q 1 , Q 2 there exists a random variable Θ such thatwhere X, X ′ , Q 1 , Q 2 , Θ are independent. In this paper we define generalized convolution of measures defined by the formula…”

mentioning

confidence: 99%

“…This is equivalent (see [11]) with the condition that for all random variables Q 1 , Q 2 there exists a random variable Θ such that…”

mentioning

confidence: 99%

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Weak stability and generalized weak convolution for random vectors and stochastic processes

Misiewicz¹

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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A random vector X is weakly stable iff for all a, b ∈ R there exists a random variable Θ such that aX + bX ′ d = XΘ. This is equivalent (see [11]) with the condition that for all random variables Q 1 , Q 2 there exists a random variable Θ such thatwhere X, X ′ , Q 1 , Q 2 , Θ are independent. In this paper we define generalized convolution of measures defined by the formulaif the equation ( * ) holds for X, Q 1 , Q 2 , Θ and µ = L(Θ). We study here basic properties of this convolution, basic properties of ⊕µ-infinitely divisible distributions, ⊕µ-stable distributions and give a series of examples.

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Section: J K Misiewiczmentioning

confidence: 99%

Section: J K Misiewiczmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…This is equivalent (see [11]) with the condition that for all random variables Q 1 , Q 2 there exists a random variable Θ such that…”

mentioning

confidence: 99%

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Weak stability and generalized weak convolution for random vectors and stochastic processes

Misiewicz¹

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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On Weak Generalized Stability of Random Variables via Functional Equations

et al. 2023

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In this paper we characterize random variables which are stable but not strictly stable in the sense of generalized convolution. We generalize the results obtained in Jarczyk and Misiewicz (J Theoret Probab 22:482-505, 2009), Misiewicz and Mazurkiewicz (J Theoret Probab 18:837-852, 2005), Oleszkiewicz (in Milman VD and Schechtman Lecture Notes in Math. 1807, Geometric Aspects of Functional Analysis (2003), Israel Seminar 2001–2002, Springer-Verlag, Berlin). The main problem was to find the solution of the following functional equation for symmetric generalized characteristic functions $$\varphi , \psi $$ φ , ψ : $$\begin{aligned}{} & {} \forall \, a,b \ge 0 \; \exists \, c(a,b) \ge 0 \; \exists \, d(a,b) \ge 0\, \forall \, t \ge 0 \\{} & {} \quad \varphi (at) \varphi (bt) = \varphi (c(a,b)t) \psi (d(a,b)t),\quad \quad \quad \quad \quad \quad \text {(A)} \end{aligned}$$ ∀ a , b ≥ 0 ∃ c ( a , b ) ≥ 0 ∃ d ( a , b ) ≥ 0 ∀ t ≥ 0 φ ( a t ) φ ( b t ) = φ ( c ( a , b ) t ) ψ ( d ( a , b ) t ) , (A) where both functions c and d are continuous, symmetric, homogeneous but unknown. We give the solution of equation (A) assuming that for fixed $$\psi , c, d$$ ψ , c , d there exist at least two different solutions of (A). To solve (A) we also determine the functions that satisfy the equation $$\begin{aligned}{} & {} \bigl (f(t(x+y)) - f(tx)\bigr ) \bigl (f(x+y) - f(y)\bigr ) \\{} & {} \quad = \bigl (f(t(x+y)) - f(ty)\bigr ) \bigl (f(x+y) - f(x)\bigr ),\quad \quad \quad \quad \quad \quad \text {(B)} \end{aligned}$$ ( f ( t ( x + y ) ) - f ( t x ) ) ( f ( x + y ) - f ( y ) ) = ( f ( t ( x + y ) ) - f ( t y ) ) ( f ( x + y ) - f ( x ) ) , (B) $$x,y,t >0$$ x , y , t > 0 , for a function $$f: (0,\infty ) \rightarrow {\mathbb {R}}$$ f : ( 0 , ∞ ) → R . As an additional result we infer that each Lebesgue measurable or Baire measurable function f satisfying equation (B) is infinitely differentiable.

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How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Jasiulis-Gołdyn,

Misiewicz,

Omey

et al. 2023

Results Math

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This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.

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Classes of measures closed under mixing and convolution. Weak stability

Cited by 22 publications

References 4 publications

Weak stability and generalized weak convolution for random vectors and stochastic processes

Weak stability and generalized weak convolution for random vectors and stochastic processes

On Weak Generalized Stability of Random Variables via Functional Equations

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

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