Strong limit theorems for a simple random walk on the 2-dimensional comb

We study the path behaviour of a simple random walk on the 2-dimensional comb lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all horizontal edges off the x-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour.


Introdution and main results
The 2-dimensional comb lattice 2 is obtained from 2 by removing all horizontal edges off the xaxis. As far as we know, the first paper that discusses the properties of a random walk on a particular tree that has the form of a comb is Weiss and Havlin [26].
A compact way of describing the just introduced transition probabilities for this simple random walk C(n) on 2 is via defining Weiss and Havlin [26] derived the asymptotic form for the probability that C(n) = (x, y) by appealing to a central limit argument. For further references along these lines we refer to Bertacchi [1].
A further insight along these lines was provided by Bertacchi [1], where she analyzed the asymptotic behaviour of the horizontal and vertical components C 1 (n), C 2 (n) of C(n) on 2 , and concluded that the expected values of various distances reached in n steps are of order n 1/4 for C 1 (n) and of order n 1/2 for C 2 (n). Moreover, this conclusion, in turn, also led her to study the asymptotic law of the random walk C(n) = (C 1 (n), C 2 (n)) on 2 via scaling the components C 1 (n), C 2 (n) by n 1/4 and n 1/2 , respectively. Namely, defining now the continuous time process C(nt) = (C 1 (nt), C 2 (nt)) by linear interpolation, Bertacchi [1] established the following remarkable weak convergence result.
where W 1 , W 2 are two independent Wiener processes (Brownian motions) and η 2 (0, t) is the local time process of W 2 at zero, and Law −→ denotes weak convergence on C([0, ∞), 2 ) endowed with the topology of uniform convergence on compact intervals.
Recall that if {W (t), t ≥ 0} is a standard Wiener process (Brownian motion), then its two-parameter local time process {η(x, t), x ∈ , t ≥ 0} can be defined via for any t ≥ 0 and Borel set A ⊂ , where λ(·) is the Lebesgue measure, and η(·, ·) is frequently referred to as Wiener or Brownian local time.
The iterated stochastic process {W 1 (η 2 (0, t)); t ≥ 0} provides an analogue of the equality in dis- where W is a standard Wiener process and X is a standard normal random variable. Namely, we have (cf., e.g., (1.7) and (1.8) in [7]) where X and Y are independent standard normal random variables.
It is of interest to note that the iterated stochastic process {W 1 (η 2 (0, t)); t ≥ 0} has first appeared in the context of studying the so-called second order limit law for additive functionals of a standard Wiener process W . See Dobrushin [13] for limiting distributions, Papanicolaou et al. [20], Ikeda and Watanabe [15], Kasahara [16] and Borodin [4] for weak convergence results.
For a related review of first and second order limit laws we refer to Csáki et al. [8], where the authors also established a strong approximation version of the just mentioned weak convergence result, and that of its simple symmetric random walk case as well, on the real line.
The investigations that are presented in this paper for the random walk C(n) on 2 were inspired by the above quoted weak limit law of Bertacchi [1] and the strong approximation methods and conclusions of Csáki et al. [7], [8], [9].
Bertacchi's method of proof for establishing the joint weak convergence statement (1.4) is based on showing that, on an appropriate probability space, each of the components converges in probability uniformly on compact intervals to the corresponding components of the conclusion of (1.4) (cf. Proposition 6.4 of Bertacchi [1]).
In this paper we extend this approach so that we provide joint strong invariance principles. Our main result is a strong approximation for the random walk C(n) = (C 1 (n), C 2 (n)) on 2 .
In this section we will now present our further results and their corollaries.
Using Theorem 1.1, we first conclude the following strong invariance principle that will enable us to establish Strassen-type functional laws of the iterated logarithm for the continuous version of the random walk process {C(x n) = (C 1 (x n), C 2 (x n)); 0 ≤ x ≤ 1} on the 2-dimensional comb lattice 2 , that is defined by linear interpolation. We have almost surely, as n → ∞, In view of this strong invariance principle we are now to study Strassen type laws of the iterated logarithm, in term of the set of limit points of the net of random vectors This will be accomplished in Theorem 1.2. In order to achieve this, we define the class (2) as the set of 2 valued, absolutely continuous functions for which f (0) = g(0) = 0,k(x)g(x) = 0 a.e., and (1.10) Theorem 1.2. Let W 1 (·) and W 2 (·) be two independent standard Wiener processes starting from 0, and let η 2 (0, ·) be the local time process at zero of W 2 (·). Then the net of random vectors as t → ∞, is almost surely relatively compact in the space C([0, 1], 2 ) and its limit points is the set of functions (2) . Corollary 1.1. For the random walk {C(n) = (C 1 (n), C 2 (n)); n = 1, 2, . . .} on the 2-dimensional comb lattice 2 we have that the sequence of random vector-valued functions is almost surely relatively compact in the space C([0, 1], 2 ) and its limit points is the set of functions (2) as in Theorem 1.2.
In order to illustrate the case of a joint functional law of the iterated logarithm for the two components of the random vectors in (1.12), we give the following example. Let where 0 ≤ K 1 ≤ K 2 ≤ 1, and we see thatk(x)g(x) = 0. Hence, provided that for A, B, K 1 and we have (k, g) ∈ (2) . Consequently, in the two extreme cases, Concerning now the joint limit points of C 1 (n) and C 2 (n), a consequence of Theorem 1.2 reads as follows.
is almost surely relatively compact in the rectangle and the set of its limit points is the domain It is of interest to find a more explicit description of D. In order to formulate the corresponding result for describing also the intrinsic nature of the domain D, we introduce the following notations: Then clearly The explicit form of A(B, K) can be easily obtained, and that of K(B) can be obtained by the solution of a cubic equation. Hence, theoretically, we have the explicit form of D 2 . However this explicit form is too complicated. A picture of D 2 can be given by numerical methods (Fig. 1.) Now we turn to investigate the liminf behaviour of the components of our random walk on 2 .
As to the liminf behaviour of the max functionals of the two components, from Bertoin [3], Nane [19] and Hirsch [14], we conclude the following Hirsch type behaviour of the respective components of the random walk process C(n) on the 2-dimensional comb lattice 2 . On account of Theorem 1.1, the so-called other law of the iterated logarithm due to Chung [6] obtains for C 2 (n).
On the other hand, for the max functional of |C 1 (·)| we do obtain a different result.
Then we have almost surely that according as to whether the integral We may construct the simple random walk on the 2-dimensional comb lattice as follows.
For any 0 < ǫ < 1 we have with probability 1 for all large enough N This and the law of large numbers for {T N } N ≥1 imply Hence, Consequently, with ǫ > 0, using increment results for the local time (cf. Csáki and Földes [10]) This completes the proof of Lemma 2.1.

Proof of Theorem 1.1.
According to the joint strong invariance principle by Révész [21], we may also assume that on our probability space we have two independent Wiener processes {W j (t); t ≥ 0} with their local time processes {η j (x, t); x ∈ , t ≥ 0}, j = 1, 2, satisfying simultaneously as n → ∞, j = 1, 2.

Proof of Theorem 1.2
The relative compactness follows from that of the components. In particular, for the first component we refer to Csáki et al. [11], while for the second one to Strassen [25]. So we only deal with the set of limit points as t → ∞.
First consider the a.s. limit points of and W 1 (η 2 (0, x t)) 2 3/4 t 1/4 (log log t) 3/4 , Clearly we need only the limit points of (3.2) but we will use the limit points of (3.1) to facilitate our argument. According to a theorem of Lévy [18], the following equality in distribution holds: Hence the set of a.s. limit points of (3.1) is the same as that of and the set of a.s. limit points of (3.2) is the same as that of By the well-known theorem of Strassen [25] the set of a.s. limit points of It follows that the set of a.s. limit points of (3.3), and hence also that of (3.1), is where Moreover, applying Theorem 3.1 of [9], we get that the set of a.s. limit points of (3.4), hence also that of (3.2), is It is easy to see thatḣ( Since ( f , ℓ) ∈ 2 , we have 1 0 (ḟ 2 (x) + (ḣ(x) −l(x)) 2 +ḣ 2 (x)) d x ≤ 1.
It remains to show that any (k, g) ∈ (2) is a limit point of (1.11).
Consider the stochastic process {V (t, ω) t ≥ 0}, ω ∈ Ω 1 , that is living on a probability space {Ω 1 , 1 , P 1 } and is equal in distribution to the absolute value of a standard Wiener process. Assume also that on this probability space there is a standard Wiener process, independent of V (·).
Our aim is now to extend this probability space so that it would carry a Wiener process W 2 (·), constructed from the just introduced stochastic process V (·). During this construction η 2 (0, ·) and W 1 (·) as well as the function k, remain unchanged, and we only have to deal with W 2 and g.
The construction will be accomplished by assigning random signs to the excursions of V (·). In order to realize this construction, we start with introducing an appropriate set of tools.
Let r(u), u ≥ 0, be a nonnegative continuous function with r(0) = 0. We introduce the following notations.
Clearly, W 2 (·) as defined in (3.9) is a standard Wiener process on the latter probability space, independent of W 1 . Moreover, V (u) = |W 2 (u)| and η 2 (0, ·) is the local time at zero of both V and W 2 . We show that (k, g) is a limit point of (1.11) in terms of W 2 that we have just defined in (3.9).
In order to accomplish the just announced goal, we first note that it suffices to consider only those g for which there are finitely many zero-free intervals (α i , β i ), i = 1, 2, . . . , m, in their support [0, 1], since the set of the latter functions is dense. Clearly then, such a function g(·) can be written as where ǫ i ∈ {−1, 1}, i = 1, . . . , m.
On account of what we just established for W 2 (·) of (3.9) above, for P 1 -almost all ω ∈ Ω 1 there exists a sequence {t with W 2 (·) as in (3.9).
On recalling the construction of the latter W 2 (·) via the excursion intervals (u ℓ j , v ℓ j ], we conclude that, for K large enough, there exists a finite number of excursion intervals (u(K, i), v(K, i)], i = 1, 2, . . . , m, such that for each ω ∈ Ω 1 for which (3.10) and the construction of the excursion intervals (u ℓ j , v ℓ j ] hold true. The finite set of the just defined intervals (u(K, i), v(K, i)] is a subset of the excursion intervals (u ℓ j , v ℓ j ] that are paired with the double sequence of i.i.d. random variables δ ℓ j in the construction of W 2 (·) as in (3.9). Let δ(K, i) denote the δ ℓ j that belongs to (u(K, i), v(K, i)). Since these random variables are independent, there exists a subsequence δ(K N , i), N = 1, 2, . . . such that we have Hence on account of (3.10) and (3.11), we have for P-almost all ω ∈ Ω.

Proof of Theorem 1.3
It is proved by Nane [19] that as u ↓ 0, with some positive constant c. Consequently, for small u we have with some positive constants c 1 and c 2 .
First assume that ∞ 1 β 2 (t)/t d t < ∞. Put t n = e n . Then we also have n β 2 (t n ) < ∞. Indeed, it is well known that the integral and series in hand are equiconvergent. For arbitrary ǫ > 0 consider the events which is summable, hence P(A n i.o.) = 0. Consequently, for large n, we have sup 0≤s≤t n and for t n ≤ t < t n+1 , we have as well Since the latter inequality is true for t large enough and ǫ > 0 is arbitrary, we arrive at lim inf t→∞ sup 0≤s≤t |W 1 (η 2 (0, s))| Put t n = e n . Hence we have also n β 2 (t n ) = ∞. Let W * (t) = sup 0≤s≤t |W 1 (η 2 (0, s))|. Consider the events consequently n P(A n ) = ∞.
Now we are to estimate P(A m A n ), (m < n). In fact, we have to estimate the probability P(W * (s) < a, W * (t) < b) for s = t m , t = t n , with a = t 1/4 m β(t m ), b = t 1/4 n β(t n ). Applying Lemma 1 of Shi [24], we have for 0 < s < t, 0 < a ≤ b, s)) .
Next we wish to estimate the expected value on the right-hand side of the latter inequality. For the sake of our calculations, we write η(0, s) instead of η 2 (0, s) to stand for the local time at zero of a standard Wiener process W (·), i.e., we also write W instead of W 2 . With this convenient notation, we now let α(s) = max{u < s : W (u) = 0} γ(s) = min{v > s : W (v) = 0}, and let g(u, v), 0 < u < s < v denote the joint density function of these two random variables. Recall that the marginal distribution of α(s) is the arcsine law with density function g 1 (u) = 1 π u(s − u) , 0 < u < s.