TY - JOUR

T1 - Drift to infinity and the strong law for subordinated random walks and Lévy processes

AU - Erickson, K. B.

AU - Maller, Ross A.

PY - 2005/4

Y1 - 2005/4

N2 - We determine conditions under which a subordinated random walk of the form S⌊ N(n)⌋ tends to infinity almost surely (a.s), or S⌊ N(n)⌋/n tends to infinity a.s., where {N(n)} is a (not necessarily integer valued) renewal process, ⌊ N(n)⌋} 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⌊ N(n)⌋ . A complication is that S⌊ N(n)⌋} 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 limn S⌊lambda n⌋/n = ∞, a.s., and lim supn S⌊lambda n⌋/n = ∞ , 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.

AB - We determine conditions under which a subordinated random walk of the form S⌊ N(n)⌋ tends to infinity almost surely (a.s), or S⌊ N(n)⌋/n tends to infinity a.s., where {N(n)} is a (not necessarily integer valued) renewal process, ⌊ N(n)⌋} 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⌊ N(n)⌋ . A complication is that S⌊ N(n)⌋} 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 limn S⌊lambda n⌋/n = ∞, a.s., and lim supn S⌊lambda n⌋/n = ∞ , 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.

KW - Drift to infinity

KW - Lévy process

KW - Random sum

KW - Random walk

KW - Strong law

KW - Subordinated process

UR - http://www.scopus.com/inward/record.url?scp=18244402889&partnerID=8YFLogxK

U2 - 10.1007/s10959-005-3507-8

DO - 10.1007/s10959-005-3507-8

M3 - Article

SN - 0894-9840

VL - 18

SP - 359

EP - 375

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 2

ER -