# zbMATH — the first resource for mathematics

Renewal equation on the whole line. (English) Zbl 0997.60096
The paper discusses the renewal equation on the whole line and proves existence of its solution provided a non-zero absolutely continuous component of a probability distribution function going in the equation. Of course, the distribution must possess non-zero (possibly infinite) mean. The presented proof is based on the theory of Volterra integral equations.

##### MSC:
 60K05 Renewal theory 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 45D05 Volterra integral equations
Full Text:
##### References:
 [1] Arabajian, L.G., Yengibarian, N.B., 1987. Convolution equations and nonlinear functional equations. J. Soviet Math. 36, 2. [2] Arabajian, L.G., 1987. On one integral equation of Radiative transfer in non-homogeneous media. Differentsial’nye Uravneniya 23(9), 1618-1622 (in Russian). [3] Blackwell, D., Extension of a renewal theorem, Pacific J. math., 3, 315-320, (1953) · Zbl 0052.14104 [4] Breiman, L., 1968. Probability Theory. Addison-Wesley, Reading, MA. · Zbl 0174.48801 [5] Feller, W., 1971. An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York. · Zbl 0219.60003 [6] Gevorgian, G.G.; Yengibarian, N.B., New theorems for the renewal integral equation., J. contemp. math. anal., 32, 1, 2-16, (1997) · Zbl 0894.45003 [7] Engibaryan, N.B.; Arutyunyan, A.A., Integral equations on the half-line with difference kernels and nonlinear functional equations, Math. USSR sb., 26, 1, 31-54, (1975) · Zbl 0333.45005 [8] Engibaryan, N.B., Convolution equations containing singular probability distributions, Izv. math., 60, 2, 251-279, (1996) · Zbl 0882.45002 [9] Engibaryan, N.B., Renewal equations on the semi-axis, Izv. math., 63, 1, 57-71, (1999) · Zbl 0937.60083 [10] Lalley, S.P., Conditional Markov renewal theory, Ann. probab., 12, 1113-1148, (1984) · Zbl 0551.60094 [11] Revuz, D., 1975. Markov Chains. North-Holland, Amsterdam. [12] Rudin, W., 1973. Functional Analysis. McGraw-Hill Book Company, New York. · Zbl 0253.46001 [13] Stone, C.J., On characteristic functions and renewal theory, Trans. amer. math. soc., 120, 327-342, (1965) · Zbl 0133.40504 [14] Stone, C.J., On absolutely continuous components and renewal theory, Ann. math. statist., 37, 271-275, (1966) · Zbl 0147.16205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.