Convergence of processes of step sums on mixing processes (Q5895276)

The standard procedure is used to construct a centred step stochastic process \[ \xi_ n(t)=n^{-1/2}\sum^{[nt]}_{k=1}(f_ n(k,x_ n(k))-E f_ n(k,x_ n(k))),\quad t\in [0,T], \] where \(f_ n(k,x)\) and \(x_ n(k)\), \(n,k=1,2,...\), are measurable functions on a measurable space (X,\({\mathcal B}_ X)\) and random variables with values X, respectively. Under the assumption that random variables satisfy the strong mixing conditions and some additional conditions it is proved that the sequence of these step processes converges in Skorokhod's topology to a stochastic process with independent increments. The weaker statement about the convergence of finite-dimensional distributions of the step processes is also proved.