Research Article | Open Access
Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks
A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.
Let be a sequence of independent and identically distributed random vectors, and write and . Let the renewal process of . A continuous time random walk (CTRW) is defined by In this setting, represents a particle jump, and is the waiting time preceding that jump, so that represents the particle location after jumps and is the time of the th jump. Then is the number of jumps by time , and the CTRW represents the particle location at time , which is a random walk subordinated to a renewal process.
It should be mentioned that the subordination scheme of CTRW processes is going back to Fogedby  and that it was expanded by Baule and Friedrich  and Magdziarz et al. . It should also be mentioned that the theory of subordination holds for nonhomogeneous CTRW processes, that were introduced in the following works: Metzler et al. [4, 5] and Barkai et al. .
The CTRW is useful in physics for modeling anomalous diffusion. Heavy-tailed particle jumps lead to superdiffusion, where a cloud of particles spreads faster than the classical Brownian motion, and heavy-tailed waiting times lead to subdiffusion. CTRW models and the associated fractional diffusion equations are important in applications to physics, hydrology, and finance; see, for example, Berkowitz et al. , Metzler and Klafter , Scalas , and Meerchaert and Scalas  for more information. In applications to hydrology, the heavy tailed particle jumps capture the velocity irregularities caused by a heterogeneous porous media, and the waiting times model particle sticking or trapping. In applications to finance, the particle jumps are price changes or log returns, separated by a random waiting time between trades.
If the jumps belong to the domain of attraction of a stable law with index , , and the waiting times belong to the domain of attraction of a stable law with index , , Becker-Kern et al.  and Meerschaert and Scheffler  showed that as , a non-Markovian limit with scaling , where is a stable Lévy motion and is the inverse or hitting time process of a stable subordinator. Densities of the CTRW scaling limit solve a space-time fractional diffusion equation that also involves a fractional time derivative of order ; see Meerschaert and Scheffler , Becker-Kern et al. , and Meerschaert and Scheffler  for complete details. Becker-Kern et al. , Meerschaert and Scheffler , and Meerschaert et al.  discussed the related limit theorems for CTRWs based on two time scales, triangular arrays and dependent jumps, respectively. The aim of the present paper is to investigate the laws of the iterated logarithm for CTRWs. We establish Chover-type laws of the iterated logarithm for CTRWs with jumps and waiting times in the domains of attraction of stable laws.
Throughout this paper we will use to denote an unspecified positive and finite constant which may be different in each occurrence and use “i.o.” to stand for “infinitely often” and “a.s." to stand for “almost surely” and “” to stand for “”. Our main results read as follows.
Theorem 1.1. Let be a sequence of i.i.d. nonnegative random variables with a common distribution , and let , independent of , be a sequence of i.i.d. nonnegative random variables with a common distribution . Assume that , , where is a slowly varying function, and that is absolutely continuous and , . Let be a sequence such that as . Then one has
The following is an immediate consequence of Theorem 1.1.
Corollary 1.2. If the tail distribution of satisfies in Theorem 1.1, then one has
In the course of our arguments we often make statements that are valid only for sufficiently large values of some index. When there is no danger of confusion, we omit explicit mention of this proviso.
2. Chung Type LIL for Stable Summands
In this section we consider a Chung-type law of the iterated logarithm for sums of random variables in the domain of attraction of a stable law, which will take a key role to show Theorem 1.1. When has a symmetric stable distribution function characterized by . Chover  established that We call (2.2) as Chover's law of the iterated logarithm. Since then, several papers have been devoted to develop Chover's LIL; see, for example, Hedye [18–20], Pakshirajan and Vasudeva , Vasudeva , Qi and Cheng , Scheffler , Chen , and Peng and Qi  for reference. For some reason the obvious corresponding statement for the “lim inf” result does not seem to have been recorded, and it is the purpose of this section to do so and may be of independent interest.
Theorem 2.1. Let be a sequence of i.i.d. nonnegative random variables with a common distribution , and let . Assume that is absolutely continuous and , , where is a slowly varying function. Then one has
In order to prove Theorem 2.1, we need some lemmas.
Lemma 2.2. Let be a slowly varying function. Then, if , , one has for any given ,
Proof. See Seneta .
Lemma 2.3. Let be a sequence of i.i.d. nonnegative random variables with a common distribution and let . Assume that is absolutely continuous and , , where is a slowly varying function. Then one has for some given small
Proof. We will follow the argument of Lemma 2.1 in Darling . Without loss of generality we can assume since each has a probability of of being the largest term, and for since is presumed continuous.
For notational simplicity we will use the tail distribution and denote by the corresponding density, so that . Then, the joint density of , given , is Thus Let us put so that It follows from Doeblin's theorem that if , for with some large . Then, for , we can choose small enough such that since has regularly varying tail distribution, so that It follows that Consider the case . By a slight transformation we find that Putting we have since and is small. Thus By (2.9) and making the change of variable to give which yields the desired result.
The following large deviation result for stable summands is due to Heyde .
Lemma 2.4. Let be a sequence of i.i.d. nonnegative random variables with a common tail distribution satisfying , , where is a slowly varying function. Let be a sequence such that as , and let be a sequence with as . Then
Now we can show Theorem 2.1.
Proof of Theorem 2.1. In order to show (2.3), it is enough to show that for all
We first show (2.18). Let , . Put again . Let be the inverse of . Obverse that , , where is a slowly varying function and , so that by Lemma 2.2. Let be i.i.d. random variables with the distribution of Uniform over , and let . Then, from the fact that is a Uniform random variable, we note that , . From (2.21), nonnegative, and and nonincreasing, it follows that Hence, the sum of the left hand side of the previously mentioned probability is finite; by the Borel-Cantelli lemma, we get Thus, by (2.20) we have Therefore, by the arbitrariness of , (2.18) holds.
We now show (2.19). Let , . For notational simplicity, we introduce the following notations: By Lemma 2.3, we have Thus, we get .
Observe again that and , so that by Lemma 2.2. Thus, we note which yields easily . Hence, since , we get . Since are independent, by the Borel-Cantelli lemma, we get
By applying Lemma 2.4 and (2.27) and some simple calculation, we have easily that , so that which, together with (2.30), implies This yields (2.19). The proof of Theorem 2.1 is now completed.
3. Proof of Theorem 1.1
Proof of Theorem 1.1. We have to show that for all
We first show (3.1). Let , . For notational simplicity, we introduce the following notations:
By (2.18), we have
Put . Let be the inverse of . Recall that , , where is a slowly varying function, so that and
Note that Thus, by noting increasing, Hence, by Lemma 2.2, Thus, by (3.8) and Lemma 2.4, we have Therefore, . By the Borel-Cantelli lemma, we get .
Observe that where stands for the complement of . Thus, letting , we have which implies that Thus, by (3.5), we have This yields (3.1) immediately by letting .
We now show (3.2). Let , . To show (3.2), it is enough to prove
By (2.19), we have
Note that Thus, by noting increasing, Hence, by Lemma 2.2, Similarly, by noting , one can have Thus, by Lemma 2.4, we have Therefore, . Since the events are independent, by the Borel-Cantelli lemma, we get .
Now, observe that Therefore, by letting , we get which implies (3.14). The proof of Theorem 1.1 is now completed.
Remark 3.1. By the proof Theorem 1.1, (1.3) can be modified as follows: That is to say that the form of (1.3) is no rare and the variables must be cut down additionally by the factors to achieve a finite lim sup.
The authors wish to express their deep gratitude to a referee for his/her valuable comments on an earlier version which improve the quality of this paper. K. S. Hwang is supported by the Korea Research Foundation Grant Funded by Korea Government (MOEHRD) (KRF-2006-353-C00004), and W. Wang is supported by NSFC Grant 11071076.
- H. C. Fogedby, “Langevin equations for continuous time Lévy flights,” Physical Review E, vol. 50, no. 2, pp. 1657–1660, 1994.
- A. Baule and R. Friedrich, “Joint probability distributions for a class of non-Markovian processes,” Physical Review E, vol. 71, no. 2, Article ID 026101, 9 pages, 2005.
- M. Magdziarz, A. Weron, and K. Weron, “Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation,” Physical Review E, vol. 75, no. 1, Article ID 016708, 2007.
- R. Metzler, E. Barkai, and J. Klafter, “Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach,” Physical Review Letters, vol. 82, no. 18, pp. 3563–3567, 1999.
- R. Metzler, J. Klafter, and I. M. Sokolov, “Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended,” Physical Review E, vol. 58, no. 2, pp. 1621–1633, 1998.
- E. Barkai, R. Metzler, and J. Klafter, “From continuous time random walks to the fractional Fokker-Planck equation,” Physical Review E, vol. 61, no. 1, pp. 132–138, 2000.
- B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, “Modeling non-fickian transport in geological formations as a continuous time random walk,” Reviews of Geophysics, vol. 44, no. 2, article RG2003, 2006.
- R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004.
- E. Scalas, “Five years of continuous-time random walks in econophysics,” in Proceedings of Workshop on Economical Heterogeneous Interacting Agents (WEHIA '04), A. Namatame, Ed., Kyoto, Japan, 2004.
- M. M. Meerschaert and E. Scalas, “Coupled continuous time random walks in finance,” Physica A, vol. 370, no. 1, pp. 114–118, 2006.
- P. Becker-Kern, M. M. Meerschaert, and H.-P. Scheffler, “Limit theorems for coupled continuous time random walks,” The Annals of Probability B, vol. 32, no. 1, pp. 730–756, 2004.
- M. M. Meerschaert and H.-P. Scheffler, “Limit theorems for continuous-time random walks with infinite mean waiting times,” Journal of Applied Probability, vol. 41, no. 3, pp. 623–638, 2004.
- M. M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2001.
- P. Becker-Kern, M. M. Meerschaert, and H.-P. Scheffler, “Limit theorem for continuous-time random walks with two time scales,” Journal of Applied Probability, vol. 41, no. 2, pp. 455–466, 2004.
- M. M. Meerschaert and H.-P. Scheffler, “Triangular array limits for continuous time random walks,” Stochastic Processes and Their Applications, vol. 118, no. 9, pp. 1606–1633, 2008.
- M. M. Meerschaert, E. Nane, and Y. Xiao, “Correlated continuous time random walks,” Statistics & Probability Letters, vol. 79, no. 9, pp. 1194–1202, 2009.
- J. Chover, “A law of the iterated logarithm for stable summands,” Proceedings of the American Mathematical Society, vol. 17, pp. 441–443, 1966.
- C. C. Hedye, “On large deviation problems for sums of random variables not attracted to the normal law,” Annals of Statistics, vol. 38, pp. 1575–1578, 1967.
- C. C. Heyde, “A contribution to the theory of large deviations for sums of independent random variables,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 7, pp. 303–308, 1967.
- C. C. Heyde, “A note concerning behaviour of iterated logarithm type,” Proceedings of the American Mathematical Society, vol. 23, pp. 85–90, 1969.
- R. P. Pakshirajan and R. Vasudeva, “A law of the iterated logarithm for stable summands,” Transactions of the American Mathematical Society, vol. 232, pp. 33–42, 1977.
- R. Vasudeva, “Chover's law of the iterated logarithm and weak convergence,” Acta Mathematica Hungarica, vol. 44, no. 3-4, pp. 215–221, 1984.
- Y. Qi and P. Cheng, “On the law of the iterated logarithm for the partial sum in the domain of attraction of stable distribution,” Chinese Annals of Mathematics A, vol. 17, pp. 195–206, 1996 (Chinese).
- H.-P. Scheffler, “A law of the iterated logarithm for heavy-tailed random vectors,” Probability Theory and Related Fields, vol. 116, no. 2, pp. 257–271, 2000.
- P. Chen, “Limiting behavior of weighted sums with stable distributions,” Statistics & Probability Letters, vol. 60, no. 4, pp. 367–375, 2002.
- L. Peng and Y. Qi, “Chover-type laws of the iterated logarithm for weighted sums,” Statistics & Probability Letters, vol. 65, no. 4, pp. 401–410, 2003.
- E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer, Berlin, Germany, 1976.
- D. A. Darling, “The influence of the maximum term in the addition of independent random variables,” Transactions of the American Mathematical Society, vol. 73, pp. 95–107, 1952.
Copyright © 2012 Kyo-Shin Hwang and Wensheng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.