On the Increments of the Brownian Sheet

Abstract

Let $\left\{ {{\text{W}}_{{\text{st}}} \;:{\text{s,t}} \in \left[ {0,1} \right]} \right\}$ be the Brownian sheet. We define the regularized process W{skst/ε} as the convolution of W_st and $\varphi _\varepsilon \left( {{\text{s,t}}} \right) = \frac{1} {{\varepsilon ^2 }}\varphi \left( {\frac{s} {\varepsilon }} \right)\varphi \left( {\frac{t} {\varepsilon }} \right)$ where ϕ is a function satisfying some conditions. For ω fixed we prove that

$$\lambda \left( {\left\{ {\left( {s,t} \right) \in \left[ {0,1} \right] \times \left[ {0,1} \right]:\frac{\varepsilon } {{\left\| \varphi \right\|_2^2 }}\frac{{\partial ^2 W_{st}^\varepsilon }} {{\partial s\partial t}} \leqslant x} \right\}} \right)\xrightarrow[{ \in \to 0}]{}\Phi \left( x \right)$$

almost surely, where λ is the Lebesgue measure in R², Φ is the standard Gaussian distribution and ‖ · ‖₂ is the usual norm in L²([− 1, 1], dx). These results are generalized to two parameter martingales M given by stochastic integrals of the Cairoli & Walsh type. Finally, as a consequence of our method we also obtain similar results for the normalized double increment of the processes W and M. These results constitute a generalisation of those obtained by Wschebor for Brownian stochastic integrals.