[For the entire collection see Zbl 0626.00026.]
Let \(S=S(t)\) and \(W=W(t)\) be random broken lines, i.e. \(S(t_ k)=X_ 1+...+X_ k\) and \(W(t_ k)=Y_ 1+...+Y_ k\) where \(0<t_ 1<...<t_ n=1\). Suppose that conditional means and conditional covariance operators of the B-valued random variables \(X_ 1,...,X_ n\) are non-random and coincide with those of the independent random variables \(Y_ 1,...,Y_ n\). In this case, under some assumptions on the Banach space B, the inequality \[ P(S\in A)-P(W\in A^ r)=C r^{- 3}\sum^{n}_{k=1}(E| X_ k|^ 3+E| Y_ k|^ 3) \] is true for any Borel set A and any real \(r>0\) where \(A^ r\) is in some sense an r-neighbourhood of the set A.
This result is proved by a special smoothing inequality which allows us to change consequently the random variables \(X_ n,X_{n-1},...,X_ 2,X_ 1\) to the variables \(Y_ n,Y_{n-1},...,Y_ 2,Y_ 1\) in the definition of the process S.