Renewal theorem for a class of stationary sequences

In an earlier paper by the author, as part of a construction of a counterexample to the central limit theorem under certain strong mixing conditions, a formula is given that shows, for strictly stationary sequences with mean zero and finite second moments and a continuous spectral density function, how that spectral density function changes if the observations in that strictly stationary sequence are "randomly spread out" in a particular way, with independent "nonnegative geometric" numbers of zeros inserted in between. In this paper, that formula will be generalized to the class of weakly stationary, mean zero, complex-valued random sequences, with arbitrary spectral measure.

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On coupling of random walks and renewal processes

Journal of Applied Probability ◽

10.2307/3215269 ◽

1996 ◽

Vol 33 (1) ◽

pp. 122-126

Author(s):

Torgny Lindvall ◽

L. C. G. Rogers

Keyword(s):

Random Walk ◽

Random Walks ◽

Simple Random Walk ◽

Renewal Processes ◽

Renewal Theorem ◽

One Dimensional ◽

Continuous State Space ◽

Continuous State ◽

Efficient Coupling ◽

Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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An elementary renewal theorem for random compact convex sets

Advances in Applied Probability ◽

10.2307/1427929 ◽

1995 ◽

Vol 27 (4) ◽

pp. 931-942 ◽

Cited By ~ 4

Author(s):

Ilya S. Molchanov ◽

Edward Omey ◽

Eugene Kozarovitzky

Keyword(s):

Support Function ◽

Convex Sets ◽

Compact Convex ◽

Renewal Function ◽

Renewal Theorem ◽

Closed Sets ◽

Random Closed Sets ◽

Minkowski Sums ◽

Image Position ◽

Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

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