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N.B. Yengibarian / Stochastic Processes and their Applications 85 (2000) 237–247
where FD; FS and FA denote, respectively, the discrete, singular and absolute contin-
uous (AC) components of F.
The problem of existence of solution of Eqs. (1.1) and (1.2) as well as existence
and determination of limits ’(ꢀ∞) are of interest in Probability Theory (see Feller,
1971; Stone, 1965,1966; Revuz, 1975; Breiman, 1968; Lalley, 1984). This problem was
studied by Stone (1966) for the case where the Fourier transform of measure dF has
a compact support. The principal result is the following Karlin’s Theorem (see Rudin,
1973, Theorem 9:15):
Theorem 1.1 (S. Karlin). Let
Z
Z
∞
∞
FA(+∞) ¿ 0;
|x| dF(x) ¡ + ∞;
ꢀ ≡ −∞ x dF(x) = 0:
(1.4)
−∞
If g ∈ L1 = L1(R); g(ꢀ∞) = 0 and ’ is a bounded function; satisfying Eqs. (1:1) and
(1:2); then the limits ’(ꢀ∞) exist and
Z
∞
’(+∞) − ’(−∞) = ꢀ−1
g(x) dx:
(1.5)
−∞
Theorem 1.1 tells nothing on solvability of Eq. (1.1).
This paper contains some new results on the existence, uniqueness and properties
of solution of Eqs. (1.1) and (1.2). The complete form of Karlin’s theorem will be
obtained: the additional condition of boundedness of g is necessary and suꢀcient for
the existence of bounded solution of Eq. (1.1). The approach is based on the author’s
method of Nonlinear Factorization Equations, on new Renewal theorem (Gevorgian
and Yengibarian, 1997) and on some facts of Tauberian Theory.
2. Notation and auxiliary propositions. One uniqueness lemma
2.1. Functional spaces
Let E be one of the following Banach spaces of the functions deÿned on R:
Lp (16p ¡+∞); M; Ml; M0; CM ; Cu; Cl; C0:
In this list M = L and CM are the spaces of essentially bounded and continu-
∞
ous bounded functions, respectively. Cu ⊂ CM is the space of uniform continuous and
bounded functions. If f ∈ Ml ⊂ M or f ∈ Cl ⊂ CM , then there exist ÿnite limits
f(ꢀ∞). If f ∈ M0 ⊂ Ml or f ∈ C0 ⊂ Cl, then f(ꢀ∞) = 0.
Let E+ = E(R+) be one of Banach spaces of the functions deÿned on R+
=
[0; ∞): LP+; M+; Ml+; M0+; CM+ ; Cl+; C0+: If f ∈ Cl+ ⊂ CM+ or f ∈ Ml+ ⊂ M+; then ∃ ÿnite
limit f(∞). If f ∈ C0+ ⊂ Cl+ or f ∈ M0+ ⊂ Ml+, then f(∞) = 0.
Denote by Lloc and Ll+oc the linear topological spaces of functions, locally integrable
on R and R+, respectively.
Let A and B are subspaces of some Linear space. Denote
A + B = {x + y; x ∈ A; y ∈ B}: