Abstract
We determine conditions under which a subordinated random walk of the form \(S_{\lfloor N(n)\rfloor}\) tends to infinity almost surely (a.s), or \(S_{\lfloor N(n)\rfloor}/n\) tends to infinity a.s., where {N(n)} is a (not necessarily integer valued) renewal process, \({\lfloor N(n)\rfloor}\) denotes the integer part of N(n), and S n is a random walk independent of {N(n)}. Thus we obtain versions of the “Alternatives”, for drift to infinity, or for divergence to infinity in the strong law, for \(S_{\lfloor N(n)\rfloor}\). A complication is that \(S_{\lfloor N(n)\rfloor}\) is not, in general, itself, a random walk. We can apply the results, for example, to the case when N(n)=λ n, λ > 0, giving conditions for lim \(_{n} S_{\lfloor \lambda n\rfloor}/n = \infty\), a.s., and lim sup \(_{n} S_{\lfloor \lambda n\rfloor}/n = \infty\), a.s., etc. For some but not all of our results, N(1) is assumed to have finite expectation. Examples show that this is necessary for the kind of behaviour we consider. The results are also shown to hold in the same degree of generality for subordinated Lévy processes.
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References
K.G. Binmore M. Katz (1968) ArticleTitleA note on the strong law of large numbers Bull. Amer. Math. Soc 74 941–943
Y.S. Chow H. Teicher (1988) Probability Theory: Independence, Interchangeability, Martingales EditionNumber2 Springer-Verlag Berlin
R. Doney (2004) ArticleTitleStochastic bounds for Lévy processes Ann. Probab. 32 1545–1552
R.A. Doney A.R. Maller (2002) ArticleTitleStability and attraction to normality for Lévy processes at zero and at infinity J. Theor. Prob 15 751–792
K.B. Erickson (1973) ArticleTitleThe strong law of large numbers when the mean is undefined Trans. Amer. Math. Soc 185 371–381
Erickson K.B., Maller R.A. (2005). Generalised Ornstein-Uhlenbeck Processes and the Convergence of Lévy Integrals Lecture Notes in Mathematics.1875, 70–94, Springer-Verlag, Berlin-Heidelberg.
W. Feller (1968) An Introduction to Probability Theory and Its Applications I EditionNumber3 Wiley Inc. New York
W. Feller (1971) An Introduction to Probability Theory and Its Applications II EditionNumber2 Wiley Inc. New York
H. Kesten (1970) ArticleTitleThe limit points of a normalized random walk Annals Math. Stat 41 1173–1205
H. Kesten R.A. Maller (1996) ArticleTitleTwo renewal theorems for random walks tending to infinity Prob. Theor. Rel. Fields 106 1–38
H. Kesten R.A. Maller (1999) ArticleTitleStability and other limit laws for exit times of random walks from a strip or a halfplane Ann. Inst. Henri Poincare 35 685–734
K. Sato (1999) Lévy Processes and Infinitely Divisible Distributions NumberInSeries68 Cambridge Studies in Advanced Mathematics. Cambridge University Press Cambridge
K. Sato (2001) Basic results on Lévy processes. O.E. Barndorff-Nielsen T. Mikosch S. Resnick (Eds) Lévy Processes Theory and Applications Birkhaüser Boston
F. Spitzer (1976) Principles of Random Walk Springer-Verlag New York
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Erickson, K.B., Maller, R.A. Drift to Infinity and the Strong Law for Subordinated Random Walks and Lévy Processes. J Theor Probab 18, 359–375 (2005). https://doi.org/10.1007/s10959-005-3507-8
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DOI: https://doi.org/10.1007/s10959-005-3507-8