Author(s): Gooty Divanji
Here we obtain almost sure limit points for a properly normalized partial sum continuous time random walk, where a continuous time random walk means a random walk subordinated to a renewal process. Continuous time random walks are used in physics to model anomalous diffusion.
Let {X n n ? 1} be a sequence of independent positive-valued stable random variables (r.v.’s), with index α, 0 < α < 1, with a common characteristic function given by
When the X n s are independent identically distributed (i.i.d.) symmetric stable r.v.’s, Chover (1966) established the law of iterated logarithm (LIL) for partial sums, by normalizing the power. He showed that
0 is the waiting time preceding that jump so that S n represents the particle location after n jumps and T n is the time of the n th jump. Then N t is the number of jumps by time t > 0, and the CTRW Yt represents the particle location at time t > 0, which is a random walk subordinated to a renewal process, as used in physics to model anomalous diffusion. CTRW modules and the associated fractional diffusion equations are important in physics, hydrology, and finance applications In applications to finance, the particle jumps are price changes or long-term trades separated by a random waiting time between trades. For more information on applications of CTRW, see the references in Hwang and Wang (2012).
Hwang and Wang (2012) and Wang (2017) obtained Chover’s form of LIL for a CTRW with jumps and waiting times in the domain of attraction of stable laws. That they established the limit supremum and limit infimum for a CTRW with jumps and waiting times in Chover’s form. The purpose of this work is to obtain a.s. limit points for a CTRW with jumps in Chover’s form for positive stable r.v.’s and to extend this to study the boundarycrossing problem.
In the next section, we present some known results. In section 3, we establish a.s. limit points for a CTRW in Chover’s form, and in the last section, we extend this to the boundary-crossing problem.
Throughout the paper, ?, C, ?, and k, with or without a suffix or super suffix, stand for positive constants, with k and n confined to being positive integers. The abbreviations i.o., a.s., and d.f. stand for infinitely often, almost surely (or almost sure) and distribution
For proof, see corollary 1.2 on page 2 of Hwang and Wang (2012) and corollary 1.2 on page 960 of Wang (2017). The following result of Hwang and Wang (2012), which is theorem 2.1 on page 3, plays a key role in establishing our result; hence, we state this result without proof.
