Asymptotic behaviour of non-isotropic random walks with heavy tails

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.


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 (∆ 1 , ∆ 2 ) with the unit vectorē = (cos θ j , sin θ j ) 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 (X j , Y j ), j ∈ N. We assume that where θ j and R j = |r j | are independent positive random variables (hereafter r.v.'s), and ∆ 1 = ∆ 2 are deterministic positive real numbers. We assume that θ j are uniformly distributed in [0, 2π) and positive r.v.'s R j are identically distributed with density f (r), r > 0. Clearly, after n steps the position reached by the moving particle is given byX A possible sample path of the random walk (2) is depicted in Figure 1 and can Fig. 1. A sample path of the random walk be interpreted as the position of a particle taking jumps at integer-valued times, with arbitrary orientation. Since ∆ 1 = ∆ 2 , the distribution of the random walk (X n ,Ŷ n ), n ≥ 1, as well as that of its asymptotic limiting process, is not rotation invariant. If the angles θ j 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 m r , 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.

Main results
In case 1) we are working under the following assumptions • (i) The jump lengths R j = |r j |, j = 1, . . . , n are exponentially distributed with parameter • (ii) The asymmetry conditions: for some C 1 , C 2 > 0 (we are interested in the case 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 R j are i.i.d. with exponential distribution whose parameter µ (n) is adjusted continuously. One can easily see that EX j = EY j = 0. Next, in view of the equality E[(cos θ) 2 ] = 1 2 . 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, Moreover, X j and Y j 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.
Theorem 1. Under the assumptions (3) and (4) the sequence (X n ,Ŷ n ) defined in (2) weakly converges to the zero-mean Gaussian vector (X, Y ) where X, Y are independent and possess the variances Our main result in Theorem 2 below establishes the limiting law for the (folded) Cauchy flights: Let us remind that the standard circular bivariate Cauchy distribution has the joint PDF (see [8], Ch.II, formula (1.19) or [9]) Theorem 2. Assume the condition (4) and fix b > 0. Let the parameter of the folded Cauchy distribution (7) for the jump lengths be scaled as a n = πb 2n . Then the distribution of random vector (X n ,Ŷ n ) weakly converges as n → ∞ to the convolution of the cumulative distribution functions F X,Y • F V,W where (X, Y ) is a zero mean Gaussian vector with independent components, and the vector (V, W ) has a circular bivariate Cauchy distribution with the shape parameter b, i.e.
Remark 1. Let us represent the position of the moving particle as a result of n random flights and n non-isotropic displacementsX n = U n + T n ,Ŷ n = V n + S n .
The expectations in the RHS of (11) may be split as n → ∞, cf. (38) below. Moreover T n and S n are asymptotically independent. However, the pair (U n , V n ) 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 J ν (x) which are solutions of ODEs and admit the expansion [1,11] Let us fix a small open neighbourhood U of (0, 0). For (α, β) ∈ U the characteristic function of the steps (X j , Y j ) reads Due to the addition formula of Bessel functions ( [4], formula 8.531, page 979) where, in our case,r 4 Proof of Theorem 1 For the sake of brevity we omit the upper index µ (n) , low index ϕ n (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 ( [4], formula 6.611, page 707).
In view of all these formulas we have that for f (r) = µe −µr ϕ(α, β) A crucial point is now to preserve only the relevant terms of the expansion of ϕ n (α, β) in view of the evaluation of the limit for the characteristic function lim n→∞ [ϕ n (α, β)] n , taking into account that µ = µ (n) , ∆ i = ∆ (n) 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 (α 2 ∆ 2 1 + β 2 ∆ 2 2 ) 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, For n → ∞, ∆ for small values of x we can obtain the following relationship We now take the equalities and thus for small values of x we have that (1 + x) −1/2 ≈ 1 − x 2 . In conclusion, by writing explicitly µ (n) , ∆ (n) i , i = 1, 2, as in the assumptions (i) and (ii) we have that (by assumptions (i) and (ii)) This concludes the proof of the Theorem 1.

Remark 2.
An interesting question is what are the implications of the assumption that ∆ 2 1 and ∆ 2 2 are negligible with respect to ∆ 1 and ∆ 2 , Let us apply the following formula ( [4], formula 6.616, page 710) By using (25), the characteristic function ϕ(α, β) can be written as Therefore, the characteristic function of (X n ,Ŷ n ), in view of the assumptions on µ (n) and ∆ (n) In the steps above we made use of µ (n) √ (µ (n) ) 2 +α 2 +β 2 ≈ 1 − α 2 +β 2 2(µ (n) ) 2 as n → ∞ in view of (22), and used assumption (i) afterwards. In contrast to (6) the terms 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 C 1 and C 2 .

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 a n , ϕ n whenever it is possible. The equalities (14) and (15) imply As in Theorem 1 all the terms of the asymptotic expansion (28) with k ≥ 2 contain the multiplyer of (α 2 ∆ 2 1 + β 2 ∆ 2 2 ) 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 The module of the term cos(kφ) in (15) is estimated by 1. Let c = α 2 + β 2 and δ be the Kronecker delta-function. Note that for any κ > 0 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 where K 1 stands for the Macdonald function [1,11]. So, we obtain that (32) If a = a n = πb 2n then for large n we have K 1 (ac) = 1 anc + anc 4 (2γ − 1) + o(n −1 ) where γ = 0.57721566 stands for the Euler-Mascheroni constant and I 1 = 1 a 2 n c 1 − a n c( 1 a n c + a n c 4 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 whereJ ν (a) stands for the Anger function which is a solution of the inhomogeneous Bessel equation Ly = (x − ν) sin πx π [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] So, we use l'Hôpital's rule to evaluate the integral I 1 (a) = lim ν→1 I ν (a) in (33) and obtain lim n→∞ lim ν→1 π a n sin(πν) J ν (a n ) − J ν (a n ) = 0.
Anyway, we need to evaluate the first term in expansion (28). According to [4], formula 6.532.6 where I 0 stands for the modified Bessel function and L 0 is the modified Struve function. Let us remind that the modified Struve function satisfies the inhomogeneous Bessel equation [1,11] Ly = x 2 d 2 y By using expansions of the modified Struve functions in a neighbourhood of 0 (see [1,11]) we write two terms of the asymptotic expansion Finally, in view of the formula 6.565 Applying (37) for ν = 0 we obtain that the characteristic function of the circular bivariate Cauchy law has the form e −b √ α 2 +β 2 . So, in the limit we obtain the product of the characteristic functions of the Gaussian law and the circular bivariate Cauchy distribution 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) r 2 +a 2 dr can be expressed in terms of the Macdonald function K 2n+1 (ac) and for even 2n the integral