INVARIANCE PRINCIPLES FOR RANKED EXCURSION LENGTHS AND HEIGHTS

In this note we prove strong invariance principles between ranked excursion lengths and heights of a simple random walk and those of a standard Brownian motion. Some consequences concerning limiting distributions and strong limit theorems will also be presented.

and heights µ i = max Clearly, the random walk does not change sign within an excursion. We may call the excursion positive (negative) if the random walk assumes positive (negative) values within this excursion. If the i-th excursion is negative, then µ + i = 0. In this paper we consider the ranked lengths and heights of excursions up to time n. In general, however the (fixed) time n need not be an excursion endpoint, and we include the length and height of this last, possibly incomplete, excursion as well. Consider the sequences and M (1) is the j-th largest in the sequence (τ 1 , τ 2 , . . . , τ ξ(n) , n − ρ ξ(n) ), M (j) (n) is the j-th largest in the sequence (µ 1 , µ 2 , . . . , µ ξ(n) , max We define M (j) the ranked lengths of the countable excursions of W over [0, t]. We mention that this sequence includes the length t − g(t) of the incomplete excursion (W (s), g(t) ≤ s ≤ t), where g(t) := sup{s ≤ t : W (s) = 0}. Let furthermore H denote the ranked heights of countable positive and all excursions, resp. of W over [0, t].
These sequences include the heights sup g(t)≤s≤t W (s) and sup g(t)≤s≤t |W (s)| of the incomplete excursion (W (s), g(t) ≤ s ≤ t).
In this paper we prove strong invariance principles for ranked lengths and heights and discuss certain consequences for limit theorems.

Invariance principle
We shall approximate the heights and lengths of random walk excursion by those of Brownian motion, using Skorokhod embedding. Define σ(0) = 0 and is a simple random walk obtained by Skorokhod embedding and we make use of the notations (ξ(n), M (j) We state below some known results as facts: Fact 2.2 Csörgő and Révész ( [7], Theorem 1.2.1) Let a t be a non-decreasing function of t such that 0 < a t ≤ t and t/a t is non-decreasing. Then lim sup  In other words, Similarly, Hence max Now, we observe that for any 0 ≤ s < t which in view of Fact 2.2 and (2.5) imply (2.2). The proof of (2.1) is similar.
Note that V j (t) − V j (s) ≤ t − s for any s ≤ t. This together with (2.5) yield (2.3), completing the whole proof of Theorem 2.1. 2

Limit theorems
It follows from our Theorem 2.1 that the limit results proved for heights and (or) lengths of excursions for the case of Brownian motion remain valid for similar quantities of simple symmetric random walk and vice versa. So the limiting distributions derived in [6] in random walk case are equivalent with the corresponding distributions in Brownian motion case.
Hence it follows from Theorem 2.1 and scaling Another form of the above distributions and further distributional results can be found in Pitman and Yor [12,13,14], and Wendel [15]. Furthermore, we mention some almost sure results proved for Brownian motion case, remaining valid also for random walk case.