Abstract
In one of his first few papers, [4], Chris gives necessary and sufficient conditions for the convergence of the series
for integers k = 0, 1, 2, …, and for a similar series with n k replaced by e rn, where r > 0. Here S n =X 1 + X 2 + … + X n is a random walk comprised of independent and identically distributed summands (X i ) i= 1, 2,…n . Series of the kind in (1) are fundamental objects in renewal theory. Under the assumptions E | X | <∞ and EX > 0 (X is any random variable having the same distribution as the X i ), Chris shows that the series in (1) converges (for the designated value of k) if and only if E | X − | k + 2 is finite, and the series with n k replaced by e rn converges for some r > 0 if and only if X − has an analytic characteristic function, or, equivalently, if Ee−θX is finite for all q in an interval [0, K], for some K > 0. (X − is defined as min(0, X).) Thus, the series in (1) converges if and only if the negative tail of the distribution of the X i is not too heavy. Two aspects are worth noting; first, the X i are allowed to take either sign (early work in renewal theory restricted S n to be a “renewal process”, i.e., comprised of non-negative summands); and, secondly, Chris obtains complete characterisations of the convergence of the series, subject only to the X i having a finite and positive expectation. Only much later was this latter restriction relaxed, in connection with (1) (in Kesten and Maller (1996)).
This research was partially supported by ARC grant DP0664603.
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Maller, R. (2010). Chris Heyde’s Work in Probability Theory, with an Emphasis on the LIL. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_4
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