MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA75 10.15559/17-MSTA75 Research Article Asymptotic behaviour of non-isotropic random walks with heavy tails KelbertMarkmkelbert@hse.rua∗ OrsingherEnzoenzo.orsingher@uniroma1.itb National Research University Higher School of Economics, Moscow RF Sapienza University of Rome Corresponding author. 2017 642017417989 24112016 932017 1432017 © 2017 The Author(s). Published by VTeX2017 Open access article under the CC BY license.

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.

 Random flights non-Gaussian limit theorem Bessel functions Introduction

We consider the problem of random flights in Euclidean spaces defined by a series of displacements, r¯j , the magnitude and direction of each one being independent of all the previous ones. This model was introduced by Karl Pearson in 1905 and has a long and interesting history, both as a purely mathematical problem in probability theory and as a model for various physical and chemical processes [2]. The majority of papers investigate the problem of random flights with the orientation of movements uniformly distributed over a sphere, and deviations separated by exponentially distributed or Dirichlet distributed time lapses (cf. discussions in [5, 6, 10]). For the most recent developments in the studying of the random walks in a random environment we refer to [3] and the papers cited therein. In this short note we introduce a novel feature in the form of non-isotropic displacements at the moments of the direction changes. As a model of this non-isotropic perturbation we consider Hadamard’s (or componentwise) product of a fixed deterministic vector (Δ1,Δ2) with the unit vector e¯=(cosθj,sinθj) in the direction of the previous movement. In the following analysis these perturbations are assumed to be small and the direction changes are frequent enough. In this note we are mainly interested in the case where the distribution of the i.i.d. flight lengths has a heavy tail, say it follows a folded Cauchy distribution.

More precisely, we consider a planar, non-isotropic random walk performed by a particle taking steps (Xj,Yj) , j∈N . We assume that Xj=(Rj+Δ1)cosθj,Yj=(Rj+Δ2)sinθj, where θj and Rj=|r¯j| are independent positive random variables (hereafter r.v.’s), and Δ1≠Δ2 are deterministic positive real numbers. We assume that θj are uniformly distributed in [0,2π) and positive r.v.’s Rj are identically distributed with density f(r) , r>0 0$]]>. Clearly, after n steps the position reached by the moving particle is given by Xˆn= ∑j=1nXj,Yˆn= ∑j=1nYj. A possible sample path of the random walk (2) is depicted in Figure 1 and can be interpreted as the position of a particle taking jumps at integer-valued times, with arbitrary orientation. Since Δ1≠Δ2 , the distribution of the random walk (Xˆn,Yˆn) , n≥1 , as well as that of its asymptotic limiting process, is not rotation invariant. If the angles θj are non-uniformly distributed on [0,2π) the resulting random motion is anisotropic as well, this case will be studied elsewhere. Here two qualitatively different examples are considered: 1) the exponential distribution of the i.i.d. flight lengths, and 2) the folded Cauchy distribution when all the moments mr , r≥1 are infinite. Naturally, in the former case under a suitable scaling one obtains the Gaussian limit with independent components having different variances. In the latter case the limiting law is a convolution of a circular bivariate Cauchy distribution with a Gaussian law.

A sample path of the random walk

 Main results

In case 1) we are working under the following assumptions

(i) The jump lengths Rj=|r¯j| , j=1,…,n are exponentially distributed with parameter μ(n)=μtn−1/2.

(ii) The asymmetry conditions: for some C1,C2>0 0$]]> (we are interested in the case C1≠C2 ) Δi(n)=Cin−1/2,i=1,2,

i.e. the displacement vectors decrease with n and are the same for all 1≤j≤n . Condition (i) means that for fixed values of n the step lengths Rj are i.i.d. with exponential distribution whose parameter μ(n) is adjusted continuously. One can easily see that EXj=EYj=0 . Next, EX12=(ER12+2Δ1ER1+Δ12)E[(cosθ)2]≈n−1(t2μ2+C1tμ+C122), in view of the equality E[(cosθ)2]=12 . In fact, (5) is an identity. Here and in what follows the symbol ≈ is used to indicate that LHS is an expansion of RHS up to O(n−k) for some k∈Z but the difference RHS–LHS is o(n−k) . Similarly, EY12≈n−1(t2μ2+C2tμ+C222). Moreover, Xj and Yj are dependent but not correlated r.v.’s. These facts suggest that a joint limiting distribution is Gaussian and is represented in the form of two independent diffusions. The proof may be provided by the standard methods via the checking Lindeberg’s conditions. However, we prefer to use a direct computation to pave the way for further results for r.v.’s with the heavy tails.

Under the assumptions (3) and (4) the sequence (Xˆn,Yˆn) defined in (2) weakly converges to the zero-mean Gaussian vector (X,Y) where X,Y are independent and possess the variances Var(X)=t2μ2+C1tμ+C122,Var(Y)=t2μ2+C2tμ+C222.

Our main result in Theorem 2 below establishes the limiting law for the (folded) Cauchy flights: f(r)=2πar2+a2,a>0,r>0. 0,r>0.\]]]> Let us remind that the standard circular bivariate Cauchy distribution has the joint PDF (see [8], Ch.II, formula (1.19) or [9]) f(x1,x2)=12π1(1+x12+x22)3/2.

Assume the condition (4) and fix b>0 0$]]>. Let the parameter of the folded Cauchy distribution (7) for the jump lengths be scaled as an=πb2n . Then the distribution of random vector (Xˆn,Yˆn) weakly converges as n→∞ to the convolution of the cumulative distribution functions FX,Y∘FV,W where (X,Y) is a zero mean Gaussian vector with independent components, Var(X)=C122,Var(Y)=C222, and the vector (V,W) has a circular bivariate Cauchy distribution with the shape parameter b, i.e. fV,W(x1,x2)=12πb(b2+x12+x22)3/2.

Let us represent the position of the moving particle as a result of n random flights and n non-isotropic displacements Xˆn=Un+Tn , Yˆn=Vn+Sn . Here Un=∑j=1nRjcosθj , Tn=Δ1∑j=1ncosθj , Vn=∑j=1nRjsinθj , Sn=Δ2∑j=1nsinθj . Then EeiαXˆn+iβYˆn=E[ei(αUn+βVn)ei(αTn+βSn)]. The expectations in the RHS of (11) may be split as n→∞ , cf. (38) below. Moreover Tn and Sn are asymptotically independent. However, the pair (Un,Vn) asymptotically follows a circular bivariate Cauchy law.

 Proofs

The initial steps are the same for both Theorems 1 and 2 and valid for any PDF f(r) of flight lengths. They are based on the properties of Bessel functions Jν(x) which are solutions of ODEs Ly=x2d2ydx2+xdydx+(x2−ν2)y=0, and admit the expansion [1, 11] Jν(x)= ∑m=0∞(−1)mm!Γ(m+k+1)(x2)2m+ν. Let us fix a small open neighbourhood U of (0,0) . For (α,β)∈U the characteristic function of the steps (Xj,Yj) reads φ(α,β)=E[exp(iα[R+Δ1]cosθ+iβ[R+Δ2]sinθ)]=12π∫02πdθ∫0∞eiα(r+Δ1)cosθ+iβ(r+Δ2)sinθf(r)dr=∫0∞J0((α2+β2)r2+2r(α2Δ1+β2Δ2)+(α2Δ12+β2Δ22))f(r)dr.

Due to the addition formula of Bessel functions ([4], formula 8.531, page 979) J0(r˜2+ρ2−2r˜ρcosϕ)=J0(r˜)J0(ρ)+2 ∑k=1∞Jk(r˜)Jk(ρ)cos(kϕ), where, in our case, r˜2=(α2+β2)r2,ρ2=(α2Δ12+β2Δ22),cosϕ=−α2Δ1+β2Δ2α2+β2α2Δ12+β2Δ22.

Proof of Theorem 1

For the sake of brevity we omit the upper index μ(n) , low index φn(α,β) , etc., whenever it is possible. We can calculate explicitly the characteristic function φ(α,β) by means of integration term by term. Further, we must keep into account the additional result ∫0∞e−αxJν(βx)dx=[α2+β2−α]νβνα2+β2,ν>−1,α>0 -1,\alpha >0\]]]> ([4], formula 6.611, page 707).

In view of all these formulas we have that for f(r)=μe−μr φ(α,β)=J0(α2Δ12+β2Δ22)∫0∞μe−μrJ0(rα2+β2)dr+2 ∑k=1∞Jk(α2Δ12+β2Δ22)cos(karcos[−α2Δ1+β2Δ2α2+β2α2Δ12+β2Δ22])×∫0∞Jk(rα2+β2)μe−μrdr=J0(α2Δ12+β2Δ22)μμ2+α2+β2+2μ ∑k=1∞Jk(α2Δ12+β2Δ22)cos(karcos[−Δ1α2+Δ2β2α2+β2α2Δ12+β2Δ22])×(α2+β2)−k/2α2+β2+μ2(μ2+α2+β2−μ)k.

A crucial point is now to preserve only the relevant terms of the expansion of φn(α,β) in view of the evaluation of the limit for the characteristic function limn→∞[φn(α,β)]n , taking into account that μ=μ(n) , Δi=Δi(n) , i=1,2 . Since (μ2+α2+β2−μ)kμ2+α2+β2=(μ2+α2+β2)k−1(1−μμ2+α2+β2)k, we can cut the terms of the expansion for k≥2 . To justify this fact let us use the expansion (13). All the terms with k≥2 in (18) contain the factors (α2Δ12+β2Δ22)k/2 and in view of assumptions (3) and (4) it is easy to check that the sum of these terms is o(n−1) uniformly over (α,β)∈U . Hence, [φn(α,β)]n≈[μα2+β2+μ2J0(α2Δ12+β2Δ22)−2J1(α2Δ12+β2Δ22)α2Δ1+β2Δ2α2+β2α2Δ12+β2Δ22×μα2+β21α2+β2+μ2(α2+β2+μ2−μ)]n=[J0(α2Δ12+β2Δ22)1+α2+β2μ2−2J1(α2Δ12+β2Δ22)(α2Δ1+β2Δ2)(α2+β2)α2Δ12+β2Δ22×μ(1−μα2+β2+μ2)]n. For n→∞ , Δi(n)→0 , i=1,2 , and since J0(x)≈1−(x2)2 as x→0 and J1(x)≈x2 for small values of x we can obtain the following relationship [φn(α,β)]n≈[(1−α2Δ12+β2Δ224)(1−α2+β22μ2)−α2Δ1+β2Δ2α2+β2μ(1−11+α2+β2μ2)]n. We now take the equalities 11+x= ∑k=0∞−1/2kxk= ∑k=0∞(−1)k2kkxk22k, and thus for small values of x we have that (1+x)−1/2≈1−x2 . In conclusion, by writing explicitly μ(n) , Δi(n) , i=1,2 , as in the assumptions (i) and (ii) we have that [φn(α,β)]n≈[(1−α2Δ12+β2Δ224)(1−α2+β22(μ(n))2)−α2Δ1+β2Δ2α2+β2μ(n)(α2+β22(μ(n))2)]n=[(1−α2Δ12+β2Δ224)(1−α2+β22(μ(n))2)−α2Δ1+β2Δ22μ(n)]n (by assumptions (i) and (ii)) ≈[1−α2C12+β2C224n−(α2+β2)t22nμ2−(α2C1n+β2C2n)t2nμ]n=(1−α2C12+β2C224n−(α2+β2)t22nμ2−α2C1+β2C22tμ1n)n→e−α2VarX+β2VarY2. This concludes the proof of the Theorem 1.

An interesting question is what are the implications of the assumption that Δ12 and Δ22 are negligible with respect to Δ1 and Δ2 , Let us apply the following formula ([4], formula 6.616, page 710) ∫0∞e−αxJ0(βx2+2γx)dx=1α2+β2eγ(α−α2+β2). By using (25), the characteristic function φ(α,β) can be written as φ(α,β)≈μμ2+α2+β2eα2Δ1+β2Δ2α2+β2(μ−μ2+α2+β2). Therefore, the characteristic function of (Xˆn,Yˆn) , in view of the assumptions on μ(n) and Δi(n) , i=1,2 , becomes [φn(α,β)]n≈[1−α2+β22(μ(n))2+α2Δ1+β2Δ2α2+β2(μ(n)−μ(n)1+α2+β2(μ(n))2)]n≈[1−α2+β22(μ(n))2−α2Δ1+β2Δ2α2+β2α2+β22μ(n)]n→e−α22(t2μ2+C1tμ)−β22(t2μ2+C2tμ). In the steps above we made use of μ(n)(μ(n))2+α2+β2≈1−α2+β22(μ(n))2 as n→∞ in view of (22), and used assumption (i) afterwards. In contrast to (6) the terms Ci22 are missing from the limiting expression (27). We conclude that the approximation considered above leads to the linearization of the limiting variances with respect to C1 and C2 .

 Proof of Theorem 2

In the case of jump lengths with a folded Cauchy distribution (7) the CLT is not applicable. Again, our goal is to calculate the limiting characteristic function keeping the relevant terms in the asymptotic expansion. We omit the lower index in an,φn whenever it is possible. The equalities (14) and (15) imply φ(α,β)=J0(α2Δ12+β2Δ22)2aπ∫0∞J0(rα2+β2)r2+a2dr−2J1(α2Δ12+β2Δ22)α2Δ1+β2Δ2α2+β2α2Δ12+β2Δ22×2aπ∫0∞J1(rα2+β2)r2+a2dr+o(n−1). As in Theorem 1 all the terms of the asymptotic expansion (28) with k≥2 contain the multiplyer of (α2Δ12+β2Δ22)k/2 and the remaining sum is o(n−1) uniformly over (α,β)∈U . Indeed, in view of (13) for k≥2 we have |Jk(α2Δ12+β2Δ22)|<Cn−k/2. The module of the term cos(kϕ) in (15) is estimated by 1. Let c=α2+β2 and δ be the Kronecker delta-function. Note that for any κ>0 0$]]> 2aπ∫0∞Jk(cr)r2+a2dr<C(n∫0n−2/k−κrk/2dr+n−1∫n−2/k−κ∞Jk(cr)r2dr)<C(n−κ−κk/2+1+κnln(n)δk=2+n−1−κk/2δk>2), 2}\Big),\end{array}\]]]> and a similar bound holds from below. Hence, the series is absolutely convergent. For these reasons it remains to consider the first two terms in the expansion (28).

For this aim note that ∫0∞J1(cr)r2+a2dr=ca2c2[∫0∞J1(u)du−∫0∞J1(u)u2u2+(ac)2du], and ∫0∞J1(u)du=−∫0∞J0′(u)du=1 . By differentiating w.r.t. the parameter the formula 6.532.4 from [4] one gets ∫0∞J1(u)u2u2+(ac)2du=acK1(ac), where K1 stands for the Macdonald function [1, 11]. So, we obtain that I1=∫0∞J1(rα2+β2)r2+a2dr=1a2(α2+β2)(1−aα2+β2K1(aα2+β2)). If a=an=πb2n then for large n we have K1(ac)=1anc+anc4(2γ−1)+o(n−1) where γ=0.57721566 stands for the Euler–Mascheroni constant and I1=1an2c[1−anc(1anc+anc4(2γ−1)+o(n−1)]=−2γ−14c+O(n−1). As a result we obtain that the second term in (28) is O(n−3/2) and does not contribute asymptotically.

Alternatively, according to [4], formula 6.532.1 for non-integer ν Iν(a)=∫0∞Jν(x)x2+a2dx=πasin(πν)(J¯ν(a)−Jν(a)), where J¯ν(a) stands for the Anger function which is a solution of the inhomogeneous Bessel equation Ly=(x−ν)sinπxπ [1]. By definition the Anger functions always coincide with the Bessel functions for the integer values of ν. The following identity is well-known [7] J¯ν(x)=sinπνπ ∑l=−∞∞(−1)lJl(x)ν−l. So, we use l’Hôpital’s rule to evaluate the integral I1(a)=limν→1Iν(a) in (33) and obtain limn→∞limν→1πansin(πν)(J¯ν(an)−Jν(an))=0.

Anyway, we need to evaluate the first term in expansion (28). According to [4], formula 6.532.6 ∫0∞J0(bx)x2+a2dx=π2a(I0(ab)−L0(ab)), where I0 stands for the modified Bessel function and L0 is the modified Struve function. Let us remind that the modified Struve function satisfies the inhomogeneous Bessel equation [1, 11] Ly=x2d2ydx2+xdydx−(x2+ν2)y=4(x/2)ν+1πΓ(ν+1/2). By using expansions of the modified Struve functions in a neighbourhood of 0 (see [1, 11]) L0(x)=(x2) ∑k=0∞1(Γ(3/2+k))2(x2)2k, we write two terms of the asymptotic expansion 2aπ∫0∞J0(rα2+β2)r2+a2dr≈1+a2(α2+β2)4−aα2+β22Γ(3/2)2+⋯

Finally, in view of the formula 6.565.3 of [4] ∫0∞xν+1(x2+a2)ν+3/2Jν(bx)dx=bνπ2νaΓ(ν+3/2)e−ab. Applying (37) for ν=0 we obtain that the characteristic function of the circular bivariate Cauchy law has the form e−bα2+β2 . So, in the limit we obtain the product of the characteristic functions of the Gaussian law and the circular bivariate Cauchy distribution limn→∞[φn(α,β)]n=exp(−C122α2−C222β2)exp(−bα2+β2), and Theorem 2 is proved.

 Acknowledgments

The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) and supported by the subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Programme.

 Appendix section

It is interesting that for odd 2n+1 the integral ∫0∞J2n+1(cr)r2+a2dr can be expressed in terms of the Macdonald function K2n+1(ac) and for even 2n the integral ∫0∞J2n(cr)r2+a2dr is a linear combination of the modified Bessel function I2n(ac) and the modified Struve functions Lk(ac) , k≤2n . Let us remind that the Macdonald function Kν(x) is a positive solution of the equation Ly=x2d2ydx2+xdydx−(x2+ν2)y=0 vanishing when x→∞ and the modified Struve function is defined in (35).

For odd order, Bessel functions ∫0∞J2n+1(cr)r2+a2dr , a,c>0 0$]]> take the form 2n+1=1:1a2c−K1(ac)a, 2n+1=3:K3(ac)a+1a2c−8a4c3, 2n+1=5:−K5(ac)a+1a2c−24a4c3+384a6c5, 2n+1=7:K7(ac)a+1a2c−48a4c3+1920a6c5−46080a8c7, 2n+1=9:−K9(ac)a+1a2c−80a4c3+5760a6c5−322560a8c7+10321920a10c9, 2n+1=11:K11(ac)a+1a2c−120a4c3+13440a6c5−1290240a8c7+92897280a10c9−3715891200a12c11. For even order, Bessel functions ∫0∞J2n(cr)r2+a2dr , a,c>0 0$]]> take the form 2n=0:πI0(ac)2a−πL0(ac)2a, 2n=2:c3−πI2(ac)2a+πL2(ac)2a, 2n=4:c15+πI4(ac)2a−πL2(ac)2a+3πL3(ac)a2c, 2n=6:a2c3105+c35−πI6(ac)2a+πL4(ac)2a−5πL3(ac)a2c+40πL4(ac)a3c2, 2n=8:a2c363+c63+πI8(ac)2a−πL4(ac)2a+12πL5(ac)a2c−84πL4(ac)a3c2+840πL5(ac)a4c3.

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