Let $\{ L^{ x}_{ t} ; (x,t)\in R^{ 1}\times R^{ 1}_{ +}\}$ denote the local time of Brownian motion and \[ \alpha_{ t}:=\int_{ -\infty}^{ \infty} (L^{ x}_{ t})^{ 2} dx . \] Let $\eta=N(0,1)$ be independent of $\alpha_{ t}$. For each fixed $t$ \[ { \int_{ -\infty}^{ \infty} (L^{ x+h}_{ t}- L^{ x}_{ t})^{ 2} dx- 4ht\over h^{ 3/2}} \stackrel{ \mathcal{ L}}{ \to}({ 64 \over 3})^{ 1/2}\sqrt{ \alpha_{ t}} \eta, \] as $h\rar 0$. Equivalently \[ { \int_{ -\infty}^{ \infty} (L^{ x+1}_{ t}- L^{ x}_{ t})^{ 2} dx- 4t\over t^{ 3/4}} \stackrel{ \mathcal{ L}}{ \to}({ 64 \over 3} )^{ 1/2}\sqrt{ \alpha_{ 1}} \eta, \] as $t\rar\infty$.
Categories: math.PR