A radial invariance principle for non-homogeneous random walks

Consider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq 2$, with the asymptotic increment covariance matrix $\sigma^2 (\mathbf{u})$ satisfying $\mathbf{u}^\top \sigma^2 (\mathbf{u}) \mathbf{u} = U$ and $\mathrm{tr}\ \sigma^2 (\mathbf{u}) = V$ in all in directions $\mathbf{u}\in\mathbb{S}^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.

We now define X = (X n , n ∈ Z + ), a discrete-time, time-homogeneous Markov process on a (non-empty, unbounded) subset X of R d . Formally, (X, B X ) is a measurable space, X is a Borel subset of R d , and B X is the σ-algebra of all B ∩ X for B a Borel set in R d . Suppose that X 0 is some fixed (i.e., non-random) point in X. Write ∆ n := X n+1 − X n for the increments of X. By assumption, given X 0 , . . . , X n , the law of ∆ n depends only on X n (and not on n); so often we ease notation by taking n = 0 and writing just ∆ for ∆ 0 . We also use the shorthand P x [ · ] = P[ · | X 0 = x] for probabilities when the walk is started from x ∈ X; similarly we use E x for the corresponding expectations.
We make the following moments assumption: The assumption (A0) ensures that ∆ has a well-defined mean vector µ(x) := E x [∆], and we suppose that the random walk has zero drift: (A1) Suppose that µ(x) = 0 for all x ∈ X.
The assumption (A0) also ensures that ∆ has a well-defined covariance matrix, which we denote by M(x) := E x [∆∆ ⊤ ], where ∆ is viewed as a column vector. To rule out pathological cases, we assume that ∆ is uniformly non-degenerate, in the following sense.
(A2) There exists v > 0 such that tr Write · op for the matrix (operator) norm given by M op = sup u∈S d−1 Mu . The following assumption on the asymptotic stability of the covariance structure of the process along rays is central.
(A3) Suppose that there exists a positive-definite matrix function σ 2 with domain S d−1 such that, as r → ∞, Finally, we assume the following.
Informally, V quantifies the total variance of the increments, while U quantifies the variance in the radial direction; necessarily U ≤ V . The final condition in (A4) is necessary to deal with the critical parameter case.
For n ∈ Z + and t ∈ R + , define For each n, we view X n as an element of the space D d := D(R + ; R d ) of functions f : Remarks. (i) As X n is typically non-Markov, Theorem 1 may be viewed as an extension of the invariance principle in [6, Thm 5.1], describing the weak convergence of a sequence of non-negative Markov processes to a Bessel diffusion.
(ii) It is well known that the stochastic differential equation (SDE) satisfied by a V -dimensional Bessel process, does not possess uniqueness in law for any V > 1 if x 0 = 0. Furthermore, if V ∈ (1, 2), uniqueness in law fails also in the case x 0 > 0 (see [2, Thm 3.2(iii)] for both assertions). Hence in the proof of Theorem 1, we work with the sequence X n 2 and show that it converges to the law BESQ V (0) of the squared Bessel process, which is uniquely determined by its SDE (see e.g. [7, Ch. XI, Sec. 1]).
(iii) Theorem 1 provides a crucial step in the proof of a full invariance principle for X n , under additional conditions. This is the subject of forthcoming work. Establishing a full invariance principle requires significantly more work, a large part of which consists of characterising the limiting diffusion that can be viewed as a generalisation of the Bessel process to many dimensions. In the present paper this work is done for us since the limit is a (squared) Bessel process.

Lemma 2.
Under assumptions (A0)-(A4), for any k ∈ N the following limits hold: where B is any compact set in R d .
The following estimates will be useful in the proof of Lemma 2.
Proof. First note that x + ∆ m 2 − x 2 = 2 x, ∆ m + ∆ m 2 . Hence by (A0) and (A1), there exists a constant C 0 > 0 such that Similarly, Then by (A0) and (A1) again, we get, for some for all m ∈ N. Taking expectations and applying (5), we find for some C 2 ∈ R + , which implies that, for some Since m x 2 ≤ m 2 + x 4 , the inequality in the lemma for ℓ = 4 follows. The case ℓ = 2 follows from (5). The remaining cases are a consequence of these bounds, the
To prove (4), take γ ∈ (0, 1/2) and observe that Hence we have from (6) that To bound the first term on the right-hand side of (7), note that chain X is a martingale. Hence, for any x ∈ X, the non-negative process X is a submartingale and Doob's L 2 inequality (see e.g. [5, Theorem 9.4]) yields For the second term on the right-hand side of (7), conditioning on X m gives by the Markov property. Then by (A0) we have that for C 1 < ∞ and all y ∈ X. It follows that The bounds in (7), (8) and (9), together with Lemma 3, show that which in turn implies (4) since γ ∈ (0, 1/2).
We need the following result from [4, Theorem 2.3]. 1{X k ∈ A} = 0, a.s. and in L q for any q ≥ 1.
Write e 1 , . . . , e d for the standard orthonormal basis vectors in R d . For convenience, set0 := e 1 .

Lemma 5.
Suppose that (A0)-(A4) hold and let k ∈ N. Then, for any linear functional φ on d × d matrices, i.e. φ : R d×d → R, the following limits in probability hold Proof. Since φ is necessarily continuous (i.e. φ op < ∞), the following estimate holds Hence, for any ε > 0, condition (A3) entails that there exists C ∈ R + such that
We now establish (12). First note that Denote by Z n the random variable in (12). By (A3), for any ε > 0 there exists a constant C < ∞ such that for all n ∈ N we have and B is defined above the display in (13). Fix X 0 = x ∈ X. Then by the ℓ = 2 case of Lemma 3, there is a constant D 1 < ∞ (depending on k) such that In order to prove Z n p −→ 0, pick arbitrary ε ′ > 0 and ε ′′ > 0, and set ε := ε ′ ε ′′ /(4D 1 ). Markov's inequality implies that Pick C < ∞ such that the inequality in (14) holds for all n ∈ N. Then, for any n ≥ 4C 2 Bk/ε ′ , the following inequalities hold: Since ε ′′ is arbitrary, we have that lim n→∞ P x [Z n > ε ′ ] = 0 and the lemma follows.
Recall that X n in (1) is a continuous-time process given in terms of the scaled Markov chain X, started at X 0 = x ∈ R d . Let Y n := X n 2 be the square of the radial component of X n . Since the square root is continuous, the mapping theorem [1, Sec. 2, Thm 2.7] implies that Theorem 1 follows if we prove that Y n converges weakly to BESQ V (0) on D 1 . This fact will be established using [3,Thm 7.4.1.,p. 354].
Let B n denote the predictable compensator of Y n . Let M n := Y n − B n be the corresponding local martingale. Define A n as the predictable compensator of the submartingale M 2 n . In particular, both A n and B n start at zero. The following proposition establishes the conditions necessary to apply [3, Thm 7.4.1., p. 354].
Proposition 6. Suppose that (A0)-(A4) hold, and that U = 1. Let T > 0. The following limits hold for any starting point X 0 = x in X: Furthermore, under P x [·], we have that sup Proof. Without loss of generality we may assume that T = 1. By definition, B n is a piece-wise constant right-continuous process started at zero with jumps at t = k/n, k ∈ {1, . . . , n}, given by using (A1), and writing B n (t−) = lim s↑t B n (s). By (A0), is a sequence of bounded random variables converging to zero point-wise. Therefore the limit in (16) follows.
We now prove the limit in (18). Note that (20) and the fact that tr M(x) = E x [ ∆ 2 ] implies that, with the usual convention that an empty sum is zero, By (A4) it holds that tr σ 2 (u) = V for all u ∈ S d−1 . Hence by (23) we find Hence for any t ∈ [0, 1] it holds that Doob's L 2 submartingale inequality and the ℓ = 2 case of Lemma 3 imply that the first term on the right-hand side of (24) converges to zero in L 1 and hence in probability. The second term converges to zero in probability by (12) in Lemma 5.
Proof of Theorem 1. As noted in Remark (ii) after the theorem, it is sufficient to prove that Y n ⇒ Y , where Y is BESQ V (0). Let g : R → R + be given by g(x) := |x| and note that Y satisfies the SDE dY t = V dt + 2g(Y t )dB t , where Y 0 = 0. It is easy to see that |g(x) − g(y)| 2 ≤ |x − y| for all x, y ∈ R. Hence pathwise uniqueness for this SDE holds for any starting point Y 0 = x 0 ∈ R by [7, Ch. IX, Thm (3.5)(ii)] (use ρ : (0, ∞) → (0, ∞), given by ρ(z) = 4z). Hence, by the Yamada-Watanabe theorem [7, Ch. IX, Thm (1.7)], the uniqueness in law holds. Thus the C 1 martingale problem for (H, δ 0 ) is well-posed, where Hf := V f ′ + 2g 2 f ′′ for any smooth f : R → R and δ 0 is the Dirac delta measure on R concentrated at zero; here C 1 denotes the space of continuous functions from R to R. Furthermore, any solution of this C 1 martingale problem has non-negative trajectories because of the support of the law of BESQ V (0) (alternatively the positivity of the paths follows from the comparison theorem [7, Ch. IX, Thm (3.7)] and the fact that BESQ 0 (0) is equal to zero at all times). Since the drift in H is constant and g 2 is continuous and non-negative on R, Proposition 6 and [3, Thm 7.4.1., p. 354] imply that Y n converges weakly to the unique solution Y of the C 1 martingale problem for (H, δ 0 ). This proves Theorem 1.