On Weak Generalized Stability of Random Variables via Functional Equations

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