Drift to Infinity and the Strong Law for Subordinated Random Walks and Lévy Processes

We determine conditions under which a subordinated random walk of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\lfloor N(n)\rfloor}$$\end{document} tends to infinity almost surely (a.s), or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\lfloor N(n)\rfloor}/n$$\end{document} tends to infinity a.s., where {N(n)} is a (not necessarily integer valued) renewal process, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lfloor N(n)\rfloor}$$\end{document} denotes the integer part of N(n), and Sn 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\lfloor N(n)\rfloor}$$\end{document}. A complication is that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\lfloor N(n)\rfloor}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{n} S_{\lfloor \lambda n\rfloor}/n = \infty$$\end{document}, a.s., and lim sup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{n} S_{\lfloor \lambda n\rfloor}/n = \infty$$\end{document}, 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.