Limit theorems for the square integral of Brownian motion and its increments
The strong laws of the iterated logarithm are proved for the integrals \[ \int^{b(T)}_{a(T)}W^ 2(t)dt, \qquad\int^{T-\alpha T}_ 0| W(t+\alpha T)-W(t)|^ 2dt \] as \(T\to\infty\), where \(1/2\leq\alpha<1\), \(W(t)\) is the standard Wiener process, \(a(T)\), \(b(T)\) are some non- decreasing functions tending to infinity. The limiting constants are calculated explicitly as solutions of some trigonometric equations.
scientific article; zbMATH DE number 90073