On Weak Generalized Stability of Random Variables via Functional Equations

Abstract

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.