Level crossings of a two-parameter random walk

We prove that the number Z ( N ) of level crossings of a two-parameter simple random walk in its first N × N steps is almost surely N 3 / 2 + o ( 1 ) as N → ∞ . The main ingredient is a strong approximation of Z ( N ) by the crossing local time of a Brownian sheet. Our result provides a useful algorithm for simulating the level sets of the Brownian sheet.