Let U1, U2,... be a sequence of independent, uniform (0, 1) r.v.'s and let R1, R2,... be the lengths of increasing runs of {Ui}, i.e., X1=R1=inf{i:Ui+1<Ui},..., Xn=R1+R2+...+Rn=inf{i:i>Xn-1,Ui+1<Ui}. The first theorem states that the sequence can be approximated by a Wiener process in strong sense. Let [tau](n) be the largest integer for which R1+R2+...+R[tau](n)[less-than-or-equals, slant]n, R*n=n-(R1+R2+...+R[tau](n)) and Mn=max{R1,R2,...,R[tau](n),R*n}. Here Mn is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of Mn. The limit distribution of the lengths of increasing runs is our third problem.
