On strong invariance for local time of partial sums (Q1068455)

The limit behavior of the local time of a random walk \((T_ n)\) generated by i.i.d. rv is studied. The local time \(\xi\) (x,n) in the point x is described as a number of crossings of the level x by the trajectory \((T_ 1,...,T_ n)\) of the random walk. The author has obtained conditions when on the same probability space one can define a Wiener process W and a random walk \((T_ n)\) such that for any \(x\in R\), \(\mu >0:\) \[ | \xi (x,n)-\eta (x,n)| =o(n^{\Delta})\quad a.s.,\quad | T_ n-W(n)| =o(n^{1/(4+\mu)})\quad a.s.\quad (n\to \infty), \] \[ and\quad \sup_{x}| \xi (x,n)-\eta (x,n)| =o(n^{1/3+\mu})\quad a.s.,\quad | T_ n-W(n)| =O(\log n)\quad a.s.\quad (n\to \infty). \] Here \(\eta\) is a local time of W, \(\Delta =\max (3/8,1/2-\mu /5(4+\mu))\).