Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $\alpha<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a conditioned process which lives on a partial compactification of the Weyl chamber.


Main results and discussion
1.1. Introduction. This note is a continuation of our paper [2]. In [2] we constructed a k-dimensional random walk conditioned to stay in the Weyl chamber of type A. The condional version of the random walk was defined via Doob's htransform. The form of the corresponding harmonic function has been suggested by Eichelsbacher and König [4]. This construction was performed under the optimal moment conditions and required the existence of k − 1 moments of the random walk.
The main aim of the present work is to consider the case, when that moment condition is not fulfilled. Instead of the existence of (k − 1)-th moment of the increment, we shall assume that the tail function is regularly varying of index 2 < α < k − 1. This assumption significantly changes the behaviour of the random walk. It turns out that the asymptotic behaviour of the exit time from the Weyl chamber depends not only on the number of walks but also on the index α. The typical sample path behaviour for the occurence of large exit times is different as Supported by the DFG. 1 well. The main reason for that is that the large exit times are caused by one (or several) big jumps of the random walk. We now introduce some notation. Let S = (S 1 , S 2 , . . . , S k ) be a k-dimensional random walk with where {X j (i)} i,j≥1 are independent copies of a random variable X. Let W denote the Weyl chamber of type A, i.e., Let τ x denote the first exit time of random walk with starting point x ∈ W , that is, τ x := min{n ≥ 1 : x + S(n) / ∈ W }. The main purpose of the present paper is to study the asymptotic behaviour of P(τ x > n) and to construct a model for ordered random walks. Recall that in order to define a random walk conditioned to stay in W , one should find a Doob h-transform E[h(x + S(1)), τ x > 1] = h(x) > 0, x ∈ W. We say that the function which satisfies the latter condition is harmonic. However, it seems that it is not possible to find a harmonic function for the Doob h-transform under present conditions. Therefore, we use a partial compactification of W , which is based on the of sample path behaviour of the random walk S on the event {τ x > n}. (Recall that a more formal way consists in applying an h-transform with a harmonic function.) Finally, we prove a functional limit theorem for random walks conditioned to stay in the Weyl chamber up to big, but finite, time.
To simplify our proofs we shall restrict our attention to the case α ∈ (k−2, k−1). However, it will be clear from the proof, that our method works also for smaller values of α.

1.2.
Tail distribution of τ x . We shall assume that EX = 0. This assumption does not restrict the generality, since τ x depends only on differences of coordinates of the random walk S. We consider a situation when increments have k − 2 finite moments, i.e., E|X| k−2 < ∞. Under this condition, for (S 1 , S 2 , . . . , S k−1 ) we can construct a harmonic function by using results of [2]. Denote this function by V (k−1) (x). It is easy to see that this function is superharmonic for our original k-dimensional random walk, i.e., and the inequality is strict at least for one This function is also superharmonic for all p, q ≥ 0.
To state our first result we introduce a convolution of v with the Green function of random walk in the Weyl chamber: for some α ∈ (k − 2, k − 1) and some k ≥ 4. Then U (x) is a strictly positive superharmonic function, i.e., E [U (x + S(1)), τ x > 1] < U (x) for all x ∈ W . Moreover, where θ is an absolute constant.
There is a very simple strategy behind formula (2). For the event {τ x > n} to occur either the random walk on the top or the random walk on the bottom should jump away, i.e., X k (l) ≈ √ n or X 1 (l) ≈ − √ n for some l ≥ 1. After such a big jump we have a system of k − 1 random walks with bounded distances between each other and one random walk on the characteristic distance √ n. This implies that the probability that all k random walks stay in W up to time n is of the same order as the probability that k − 1 random walk stay in W up to time n − l. But it follows from (1) that E[|X| k−2 ] < ∞. So we can apply Theorem 1 from [2], which says that the latter probability is of order n −(k−1)(k−2)/4 . Since P(|X| > √ n) ∼ n −α/2 , we see that P(τ x > n) is of order n −α/2−(k−1)(k−2)/4 . This strategy sheds also some light on the structure of the function U (x): the l-th summand in the series corresponds to the case, when big jump occurs at time l + 1.

1.3.
Construction of a conditioned random walk. Since U is not harmonic, we can not use the Doob h-transform with this function to define a random walk, conditioned to stay in W for all times. (More precisely, an h-transform with a superharmonic function leads to strict substohastic transition kernel.) An alternative approach via distributional limit does not work as well: using Theorem 1 we can defineP (x, A) for any x ∈ W and for any bounded A ⊂ W by the relation Then we can extendP (x, ·) to a finite measure on the Borel subsets of W . But this measure is not probabilistic, sincê We loose the mass because of an "infinite" jump in the first step. Indeed, according to the optimal strategy in Theorem 1, one of the random walks should have a jump of order n 1/2 , and we let n go to infinity. This infinite jump is the reason, why a Markov chain, corresponding to the kernelP (x, A) has almost sure finite lifetime. Similar effects have been observed already in other models. Bertoin and Doney [1] have proven that a one-dimensional random walk with negative drift and regularly varying tail conditioned to stay positive has finite lifetime. Jacka and Warren [8] have shown that the same effect appears in the Kolmogorov K2 chain.
The asymptotic behaviour of the corresponding Markov chain, say {Ŝ(n), n ≥ 0}, can be described as follows. One of the random walks jumps away at time m with probability E [v(x + S(m − 1)), τ x > m − 1] /U (x). Then we restart our process, which has from now on one frozen coordinate, either −∞ or ∞, and k − 1 ordered random walks. But for k − 1 random walks we can apply Theorem 3 of [2]. As a result we have that the limit of Ŝ ([nt])/ √ n, t ∈ [r n /n, 1] converges weakly to a process {X(t), t ∈ (0, 1]}, where r n is such that r n → ∞. (We need this additional restriction because of jumps at bounded times.) The limit can be constructed as follows: Let D(t) denote here the (k − 1)-dimensional Dyson Brownian motion starting from zero. With some probability p(x) we add to D(t) one coordinate with constant value ∞, and with probability q(x) = 1 − p(x) we add the coordinate with value −∞.
We have constructed a model of ordered random walks on an enlarged state space by formalising an intuitive picture of big jumps. But it remains unclear whether one can find a harmonic function for the substochastic kernel P(x + S(1) ∈ dy, τ x > 1). If such a function exists, then one can construct a model of ordered random walks on the original Weyl chamber. We conjecture that there are no harmonic functions for ordered random walks with heavy tails. The examples from [1,8], which we have mentioned above, support this conjecture.
The most standard way to describe the set of harmonic functions consists in the study of the corresponding Martin boundary. We found only a few results on Martin boundary for killed random walks. Doney [3] found sufficient and necessary conditions for existence of harmonic functions in one-dimensional case. The proof relies on the Wiener-Hopf factorisation, which seems to work in the one-dimensional case only. In a series of papers [5,6,7] by Ignatiouk-Robert, and by Ignatiouk-Robert and Loree Martin boundaries for killed random walks with non-zero drift in a half-space and in a quadrant have been studied. In all these papers the Cramer condition has been imposed. Next-neighbour random walks with zero mean in the Weyl chamber have been studied by Raschel [11,12]. In our situation all the increments are heavy-tailed. This means that one needs another method for finding the Martin boundary. We define Here r n → ∞ and r n = o(n). (Again, we need to go away from zero, because of a big jump occurring at the very beginning.) In order to state our limit theorem we have to introduce a limiting process, say X. The distribution of the starting point, X(0), is given by Further, given X(0) = y, we define where y(a) = y + a(0, 1, 2, . . . , k − 1).

Theorem 2.
Under the conditions of Theorem 1, It is worth mentioning that the limiting process is not invariant with respect to the starting position of the random walk. More precisely, the distribution of X(0) depends on x through p(x) and q(x). Clearly this happens beacuse of one large jump i the beginning. An analogous result can be proven also for random walks with E|X| k−1 < ∞, but the limiting process will start always at zero.
1.5. Some remarks on the general case. Although the informal picture behind Theorems 1 and 2 is quite simple, the proofs are very technical. In the case of smaller values of α, i.e. α < k − 2, one has to overcome even more technical difficulties, which are of the combinatorial nature. However, is it clear that our approach works in the case α < k − 2 as well. In this paragraph we describe the behaviour of ordered random walks for such values of α.
First, in order to stay in W at least up to time n, the random walk S should have k α := k − [α + 1] big jumps. Then it may happen that at least two jumps go in the same direction (upwards or downwards). The values of all these jumps should be ordered. As a result one gets the following relation: with some superharmonic function U . Second, to construct ordered random walks we need to add all vectors with k α infinite coordinates. Finally, in Theorem 2 one has to change the distribution of X 0 only: The limiting process will start from a random point with k α non-zero coordinates.
Unfortunately, the case of integer values of α remains unsolved. If, for example, α = k − 1, then, the jumps of order √ n do not contribute to P(τ x > n). Therefore, we can not use the method proposed in the present work.
If α < 2, then one can describe the asymptotic behaviour of τ x . If k = 2, then P(τ x > n) is of order n −1/2 . And if k ≥ 3, then we expect k − 2 big jumps. But in this situation all big jumps are of order n 1/α . As a result one will obtain Our last remark concerns other Weyl chambers. König and Schmid [9] have shown that the approach proposed in [2] works also in Weyl chambers of types C and D. It is easy to see that, using the method from the present paper, one can prove analogons of Theorems 1 and 2 for chambers of types C and D. Moreover, since big negative jumps lead to exit from these two regions, the corresponding optimal strategies are even simpler then in the chamber of type A.

Finiteness of the superharmonic function
We first introduce some notation. For every ε > 0 denote and let ν n := min{j ≥ 1 : x + S(j) ∈ W n,ε } be the first time the random walk enters this region.
Proof. Fix δ > 0. Let η ± be the moments of 'big' jumps upwards and downwards, i.e., First we note that To estimate the second term we apply Proposition 4 of [2] to obtain Then, according to Lemma 8 in [2], This gives us an estimate for the second term of (3). The rest of the proof is devoted to estimation of the first summand in (3). We split this term in three parts: with big jump upwards, big jump downwards and no big jumps, We construct estimates for each of terms separately and then combine them. We apply the resulting estimate recursively several times and prove the claim.
Big jump upwards: Using the Markov property, we get We apply Proposition 4 of [2] to the system of k − 1 random walks, Therefore, E[v 1 (y + S(n − l)), τ y > n − l] ≤ Cv 1 (y) and, consequently, Using the Markov property once again, we have The random variable X k is independent of X 1 , . . . , X k−1 , Hence, Summing up over l ≤ n 1−ε , we obtain Big jump downwards: We now turn our attention to the case when all jumps of the random walk on the top are bounded by n (1−δ)/2 . First of all we note that according to one of Fuk-Nagaev inequalities, see Corollary 1.11 in [10], where r(δ) = δ/2 − δ 2 /2. This yields Next we need to analyse the case when the top random walk is always less than n 1/2 . Hence, Clearly in the definition of E down,l the big jum occurs at time l. Also note that we have excluded the possibility that the top random walk goes up without a big jump. Applying the Markov property again, Using the Markov property for the multiplier, E af ter,l (y) = Wn,ε P y + S(r) ∈ dz, τ y > r = ν n , max j≤r S k (j) < n 1/2 It follows from the martingale property of v 1 that Consequently, It follows from Proposition 4 of [2] that, uniformly in z ∈ W n,ε , Therefore, since S νn ∈ W n,ε and S k (ν n ) < n 1/2 it follows from the latter inequality and (8) that E af ter,l (y) where M 1 (n) := min k≤n S 1 (k). Using now the fact that the sequence v 2 (y + S n )1{τ (2) y > n} is a martingale, we get E af ter,l (y) Applying the Rosenthal inequality, we get finally E af ter,l (y) ≤ Cn −(1/2−ε)(k−2) v 2 (y) |y 1 | k−2 + n (k−2)/2 .

Proof of Theorem 1
We start by estimating the tail of τ x for paths without big jumps.
Applying now estimate (33) from [2], we get Consequently, Repeating arguments from the proof of Proposition 3, we get As a result we have Analogously, Therefore, it remains to consider the case when η + ≤ ν n and η − ≤ ν n . Because of the symmetry we may assume that η + ≤ η − . Then P τ x > n, η + = l, η − = j .

First we note
whereṽ is the invariant function for random walks (S 2 , . . . , S k−1 ). Using now the bound we obtain Furthermore, applying (21) once again, we get
Lemma 5. If S is as in Theorem 1 of [2], then there exists a constant C such that Proof. It follows from Proposition 4 of [2] that V (x) ∼ ∆(x) uniformly in x ∈ W n,ε . This and inequality (33) from [2] imply that Recalling that the sequence V (x + S(n))1{τ x > n} is a martingale, we conclude that To complete the proof it remains to note that P(ν n > n 1−ε ) ≤ e −Cn ε and that inf x∈W V (x) > 0. Lemma 6. If x k = r √ n and x 1 , . . . , x k−1 are fixed, then there exists a function ψ such that Moreover, ψ(a) ≤ Ca k−1 , a > 0.