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.

1. Introduction

Let {Y_i, J_i} be a sequence of independent and identically distributed random vectors, and write S(n) = Y₁ + Y₂ + ⋯+Y_n and T(n) = J₁ + J₂ + ⋯+J_n. Let N_t = max {n ≥ 0 : T(n) ≤ t} the renewal process of J_i. A continuous time random walk (CTRW) is defined by

(1.1)

In this setting, Y_i represents a particle jump, and J_i > 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 nth jump. Then N_t is the number of jumps by time t > 0, and the CTRW X(t) represents the particle location at time t > 0, 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 [1] and that it was expanded by Baule and Friedrich [2] and Magdziarz et al. [3]. 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. [6].

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. [7], Metzler and Klafter [8], Scalas [9], and Meerchaert and Scalas [10] 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 Y_i belong to the domain of attraction of a stable law with index α, (0 < α < 2), and the waiting times J_i belong to the domain of attraction of a stable law with index β, (0 < β < 1), Becker-Kern et al. [11] and Meerschaert and Scheffler [12] showed that as c → ∞,

(1.2)

a non-Markovian limit with scaling

, where A(t) is a stable Lévy motion and E(t) is the inverse or hitting time process of a stable subordinator. Densities of the CTRW scaling limit A(E(t)) solve a space-time fractional diffusion equation that also involves a fractional time derivative of order β; see Meerschaert and Scheffler [13], Becker-Kern et al. [11], and Meerschaert and Scheffler [12] for complete details. Becker-Kern et al. [14], Meerschaert and Scheffler [15], and Meerschaert et al. [16] 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 C 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 “u(x) ~ v(x)” to stand for “lim u(x)/v(x) = 1”. Our main results read as follows.

Theorem 1.1. Let {Y_i} be a sequence of i.i.d. nonnegative random variables with a common distribution F, and let {J_i}, independent of {Y_i}, be a sequence of i.i.d. nonnegative random variables with a common distribution G. Assume that 1 − F(x) ~ x^−αL(x), 0 < α < 2, where L is a slowly varying function, and that G is absolutely continuous and 1 − G(x) ~ Cx^−β, 0 < β < 1. Let {B(n)} be a sequence such that nL(B(n))/B(n) ^α → C as n → ∞. Then one has

(1.3)

The following is an immediate consequence of Theorem 1.1.

Corollary 1.2. If the tail distribution of Y_i satisfies P(Y₁ > x) ~ Cx^−α in Theorem 1.1, then one has

(1.4)

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 J_i has a symmetric stable distribution function G characterized by

(2.1)

0 < β < 2. Chover [17] established that

(2.2)

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 [21], Vasudeva [22], Qi and Cheng [23], Scheffler [24], Chen [25], and Peng and Qi [26] 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 {J_i} be a sequence of i.i.d. nonnegative random variables with a common distribution G(x), and let V(x) = inf {y > 0 : 1 − G(y) ≤ 1/x}. Assume that G is absolutely continuous and 1 − G(x) ~ x^−βl(x), 0 < β < 1, where l is a slowly varying function. Then one has

(2.3)

In order to prove Theorem 2.1, we need some lemmas.

Lemma 2.2. Let h(x) be a slowly varying function. Then, if y_n → ∞, z_n → ∞, one has for any given τ > 0,

(2.4)

Proof. See Seneta [27].

Lemma 2.3. Let {J_i} be a sequence of i.i.d. nonnegative random variables with a common distribution G and let M(n) = max {J₁, J₂, …, J_n}. Assume that G is absolutely continuous and 1 − G(x) ~ x^−βl(x), 0 < β < 1, where l is a slowly varying function. Then one has for some given small t > 0

(2.5)

Proof. We will follow the argument of Lemma 2.1 in Darling [28]. Without loss of generality we can assume J₁ = max {J₁, J₂, …, J_n} = M(n) since each J_i has a probability of 1/n of being the largest term, and P(J_i = J_j) = 0 for i ≠ j since G(x) is presumed continuous.

For notational simplicity we will use the tail distribution and denote by g(x) the corresponding density, so that . Then, the joint density of J₁, J₂, …, J_n, given J₁ = M(n), is

(2.6)

Thus

(2.7)

Let us put

(2.8)

so that

(2.9)

It follows from Doeblin′s theorem that if λ > 0,

(2.10)

for y ≥ y₀ with some large y₀ > 0. Then, for y ≤ y₀, we can choose t > 0 small enough such that t < −log G(y₀) since G has regularly varying tail distribution, so that

(2.11)

It follows that

(2.12)

Consider the case y ≥ y₀. By a slight transformation we find that

(2.13)

Putting

(2.14)

we have η < 1 since 0 < β < 1 and t is small. Thus

(2.15)

By (2.9) and making the change of variable

to give

(2.16)

which yields the desired result.

The following large deviation result for stable summands is due to Heyde [19].

Lemma 2.4. Let {ξ_i} be a sequence of i.i.d. nonnegative random variables with a common tail distribution satisfying P(ξ₁ > x) ~ x^−rh(x), 0 < r < 2, where h is a slowly varying function. Let {λ_n} be a sequence such that as n → ∞, and let {x_n} be a sequence with x_n → ∞ as n → ∞. Then

(2.17)

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 ɛ > 0

(2.18)

(2.19)

We first show (2.18). Let n_k = [θ^k], 1 < θ < 2. Put again . Let be the inverse of . Obverse that , 0 < y ≤ 1, where H is a slowly varying function and , so that

(2.20)

(2.21)

by Lemma 2.2. Let U, U₁, U₂, …, U_n be i.i.d. random variables with the distribution of U Uniform over (0,1), and let M^*(n) = max {U₁, U₂, …, U_n}. Then, from the fact that G(J_n) is a Uniform (0,1) random variable, we note that

, n ≥ 1. From (2.21), J_i nonnegative, and

and

nonincreasing, it follows that

(2.22)

Hence, the sum of the left hand side of the previously mentioned probability is finite; by the Borel-Cantelli lemma, we get

(2.23)

Thus, by (2.20) we have

(2.24)

Therefore, by the arbitrariness of θ > 1, (2.18) holds.

We now show (2.19). Let , δ > 0. For notational simplicity, we introduce the following notations:

(2.25)

By Lemma 2.3, we have

(2.26)

Thus, we get ∑ P(O_k) < ∞.

Observe again that and V(n) ~ n^1/βH(n), so that

(2.27)

(2.28)

by Lemma 2.2. Thus, we note

(2.29)

which yields easily

. Hence, since P(E_k) ≥ P(F_k) − P(O_k), we get ∑ P(E_k) = ∞. Since E_k are independent, by the Borel-Cantelli lemma, we get

(2.30)

By applying Lemma 2.4 and (2.27) and some simple calculation, we have easily that , so that

(2.31)

which, together with (2.30), implies

(2.32)

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 ɛ > 0

(3.1)

(3.2)

We first show (3.1). Let t_k = θ^k, 1 < θ < 2. For notational simplicity, we introduce the following notations:

(3.3)

By (2.18), we have

(3.4)

Put . Let be the inverse of . Recall that , 0 < y ≤ 1, where is a slowly varying function, so that and

(3.5)

Note that

(3.6)

Thus, by noting U increasing,

(3.7)

Hence, by Lemma 2.2,

(3.8)

Thus, by (3.8) and Lemma 2.4, we have

(3.9)

Therefore,

. By the Borel-Cantelli lemma, we get

Observe that

(3.10)

where E^c stands for the complement of E. Thus, letting n → ∞, we have

(3.11)

which implies that

(3.12)

Thus, by (3.5), we have

(3.13)

This yields (3.1) immediately by letting θ ↓ 1.

We now show (3.2). Let , δ > 0. To show (3.2), it is enough to prove

(3.14)

Put

(3.15)

By (2.19), we have

(3.16)

Note that

(3.17)

Thus, by noting U₁ increasing,

(3.18)

Hence, by Lemma 2.2,

(3.19)

Similarly, by noting t_k/t_k−1 → ∞, one can have

(3.20)

Thus, by Lemma 2.4, we have

(3.21)

Therefore, ∑ P(W_k) = ∞. Since the events {W_k} are independent, by the Borel-Cantelli lemma, we get P(W_k i.o.) = 1.

Now, observe that

(3.22)

Therefore, by letting m → ∞, we get

(3.23)

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:

(3.24)

That is to say that the form of (1.3) is no rare and the variables (B(t^β)) ⁻¹X(t) must be cut down additionally by the factors (log t) ^−1/α to achieve a finite lim sup.

Acknowledgments

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.

References

1 Fogedby H. C., Langevin equations for continuous time Lévy flights, Physical Review E. (1994) 50, no. 2, 1657–1660, 2-s2.0-4244190920, https://doi.org/10.1103/PhysRevE.50.1657.
10.1103/PhysRevE.50.1657
CAS PubMed Web of Science® Google Scholar
2 Baule A. and Friedrich R., Joint probability distributions for a class of non-Markovian processes, Physical Review E. (2005) 71, no. 2, 9, 026101, https://doi.org/10.1103/PhysRevE.71.026101, 2139958.
10.1103/PhysRevE.71.026101
CAS Web of Science® Google Scholar
3 Magdziarz M., Weron A., and Weron K., Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation, Physical Review E. (2007) 75, no. 1, 2-s2.0-33846621507, https://doi.org/10.1103/PhysRevE.75.016708, 016708.
10.1103/PhysRevE.75.016708
CAS PubMed Web of Science® Google Scholar
4 Metzler R., Barkai E., and Klafter J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Physical Review Letters. (1999) 82, no. 18, 3563–3567, 2-s2.0-18144408075.
10.1103/PhysRevLett.82.3563
CAS Web of Science® Google Scholar
5 Metzler R., Klafter J., and Sokolov I. M., Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended, Physical Review E. (1998) 58, no. 2, 1621–1633, 2-s2.0-0001536558.
10.1103/PhysRevE.58.1621
CAS Web of Science® Google Scholar
6 Barkai E., Metzler R., and Klafter J., From continuous time random walks to the fractional Fokker-Planck equation, Physical Review E. (2000) 61, no. 1, 132–138, https://doi.org/10.1103/PhysRevE.61.132, 1736459.
10.1103/PhysRevE.61.132
CAS Web of Science® Google Scholar
7 Berkowitz B., Cortis A., Dentz M., and Scher H., Modeling non-fickian transport in geological formations as a continuous time random walk, Reviews of Geophysics. (2006) 44, no. 2, article RG2003, 2-s2.0-33845247896, https://doi.org/10.1029/2005RG000178.
10.1029/2005RG000178
Web of Science® Google Scholar
8 Metzler R. and Klafter J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A. (2004) 37, no. 31, R161–R208, https://doi.org/10.1088/0305%2D4470/37/31/R01, 2090004, ZBL1075.82018.
10.1088/0305-4470/37/31/R01
Web of Science® Google Scholar
9 Scalas E., A. Namatame, Five years of continuous-time random walks in econophysics, Proceedings of Workshop on Economical Heterogeneous Interacting Agents (WEHIA ′04), 2004, Kyoto, Japan.
Google Scholar
10 Meerschaert M. M. and Scalas E., Coupled continuous time random walks in finance, Physica A. (2006) 370, no. 1, 114–118, https://doi.org/10.1016/j.physa.2006.04.034, 2263769.
10.1016/j.physa.2006.04.034
Web of Science® Google Scholar
11 Becker-Kern P., Meerschaert M. M., and Scheffler H.-P., Limit theorems for coupled continuous time random walks, The Annals of Probability B. (2004) 32, no. 1, 730–756, https://doi.org/10.1214/aop/1079021462, 2039941, ZBL1054.60052.
10.1214/aop/1079021462
Web of Science® Google Scholar
12 Meerschaert M. M. and Scheffler H.-P., Limit theorems for continuous-time random walks with infinite mean waiting times, Journal of Applied Probability. (2004) 41, no. 3, 623–638, 2074812, https://doi.org/10.1239/jap/1091543414, ZBL1065.60042.
10.1239/jap/1091543414
Web of Science® Google Scholar
13 Meerschaert M. M. and Scheffler H.-P., Limit Distributions for Sums of Independent Random Vectors, 2001, John Wiley & Sons, New York, NY, USA, Wiley Series in Probability and Statistics: Probability and Statistics, 1840531.
Google Scholar
14 Becker-Kern P., Meerschaert M. M., and Scheffler H.-P., Limit theorem for continuous-time random walks with two time scales, Journal of Applied Probability. (2004) 41, no. 2, 455–466, 2052584, https://doi.org/10.1239/jap/1082999078, ZBL1050.60038.
10.1239/jap/1082999078
Web of Science® Google Scholar
15 Meerschaert M. M. and Scheffler H.-P., Triangular array limits for continuous time random walks, Stochastic Processes and Their Applications. (2008) 118, no. 9, 1606–1633, https://doi.org/10.1016/j.spa.2007.10.005, 2442372, ZBL1153.60023.
10.1016/j.spa.2007.10.005
Web of Science® Google Scholar
16 Meerschaert M. M., Nane E., and Xiao Y., Correlated continuous time random walks, Statistics & Probability Letters. (2009) 79, no. 9, 1194–1202, https://doi.org/10.1016/j.spl.2009.01.007, 2519002, ZBL1179.60024.
10.1016/j.spl.2009.01.007
Web of Science® Google Scholar
17 Chover J., A law of the iterated logarithm for stable summands, Proceedings of the American Mathematical Society. (1966) 17, 441–443, 0189096, https://doi.org/10.1090/S0002%2D9939%2D1966%2D0189096%2D2, ZBL0144.40503.
10.1090/S0002-9939-1966-0189096-2
Web of Science® Google Scholar
18 Hedye C. C., On large deviation problems for sums of random variables not attracted to the normal law, Annals of Statistics. (1967) 38, 1575–1578.
10.1214/aoms/1177698712
Web of Science® Google Scholar
19 Heyde C. C., A contribution to the theory of large deviations for sums of independent random variables, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. (1967) 7, 303–308, 0216549, ZBL0158.17001.
10.1007/BF00535016
Web of Science® Google Scholar
20 Heyde C. C., A note concerning behaviour of iterated logarithm type, Proceedings of the American Mathematical Society. (1969) 23, 85–90, 0251772, https://doi.org/10.1090/S0002%2D9939%2D1969%2D0251772%2D3, ZBL0185.46901.
10.1090/S0002-9939-1969-0251772-3
Web of Science® Google Scholar
21 Pakshirajan R. P. and Vasudeva R., A law of the iterated logarithm for stable summands, Transactions of the American Mathematical Society. (1977) 232, 33–42, 0455093, https://doi.org/10.1090/S0002%2D9947%2D1977%2D0455093%2D4, ZBL0325.60034.
10.1090/S0002-9947-1977-0455093-4
Web of Science® Google Scholar
22 Vasudeva R., Chover′s law of the iterated logarithm and weak convergence, Acta Mathematica Hungarica. (1984) 44, no. 3-4, 215–221, https://doi.org/10.1007/BF01950273, 764612, ZBL0555.60022.
10.1007/BF01950273
Web of Science® Google Scholar
23 Qi Y. and Cheng P., On the law of the iterated logarithm for the partial sum in the domain of attraction of stable distribution, Chinese Annals of Mathematics A. (1996) 17, 195–206.
Google Scholar
24 Scheffler H.-P., A law of the iterated logarithm for heavy-tailed random vectors, Probability Theory and Related Fields. (2000) 116, no. 2, 257–271, https://doi.org/10.1007/PL00008729, 1743772, ZBL0964.60034.
10.1007/PL00008729
Web of Science® Google Scholar
25 Chen P., Limiting behavior of weighted sums with stable distributions, Statistics & Probability Letters. (2002) 60, no. 4, 367–375, https://doi.org/10.1016/S0167%2D7152(02)00286%2D9, 1947176, ZBL1014.60010.
10.1016/S0167-7152(02)00286-9
Web of Science® Google Scholar
26 Peng L. and Qi Y., Chover-type laws of the iterated logarithm for weighted sums, Statistics & Probability Letters. (2003) 65, no. 4, 401–410, https://doi.org/10.1016/j.spl.2003.08.009, 2039884, ZBL1116.60323.
10.1016/j.spl.2003.08.009
Web of Science® Google Scholar
27 Seneta E., Regularly Varying Functions, 1976, Springer, Berlin, Germany, Lecture Notes in Mathematics, Vol. 508, 0453936.
10.1007/BFb0079658
Google Scholar
28 Darling D. A., The influence of the maximum term in the addition of independent random variables, Transactions of the American Mathematical Society. (1952) 73, 95–107, 0048726, https://doi.org/10.1090/S0002%2D9947%2D1952%2D0048726%2D0, ZBL0047.37502.
10.1090/S0002-9947-1952-0048726-0
Web of Science® Google Scholar

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Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

Abstract

1. Introduction

2. Chung Type LIL for Stable Summands

3. Proof of Theorem 1.1

Acknowledgments

References

References

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Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

Abstract

1. Introduction

2. Chung Type LIL for Stable Summands

3. Proof of Theorem 1.1

Acknowledgments

References

References

Related

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