The Range of a Simple Random Walk on $\mathbb{Z}$: An Elementary Combinatorial Approach

Abstract

Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on $\mathbb{Z}$ of length $n$ are presented. Both of them rely on Hermann Weyl's discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on $\mathbb{Z}$ can be turned into a known path-enumeration problem on a bounded lattice. The solution is provided by means of the adjacency matrix $\mathbf Q_d$ of the walk on a bounded lattice $(0,1,\ldots,d)$. The second approach is algebraic in nature, and starts with the adjacency matrix $\mathbf{Q_d}$. The powers of the adjacency matrix are expanded in terms of products of non-commutative left and right shift matrices. The representation of such products by means of the discrepancy norm reveals the solution directly.

Extreme order statistics of random walks

“…Consequently, EW n,n / √ n → 8/π and Var W n,n /n → 4(ln 2 − 2/π) as n → ∞. The exact distribution of W n,n for a simple random walk, with increments ±1 was derived in [43,66], but the formula is complicated. For α < 1, the limit distribution of W ⌊αn⌋,n is more involved than that of W n,n , and deriving explicit formulas seems to be far-fetched.…”

On Quasi-Isometry of Threshold-Based Sampling

“…D . Based on the concept of minimal length intervals of maximal discrepancy (MMD intervals), introduced in [43], we will show Proposition 5.1.…”

How Far Might We Walk at Random?

“…are unbounded. These results have implications for the range M + n + M − n of a random walk [14,15,16]. More will be said about M − n moments in Section 3.…”