Let a=(ai)i=1∞ be a strictly increasing sequence of natural numbers and let 𝒜 be a space of Lebesgue measurable functions defined on [0,1). Let {y} denote the fractional part of the real number y. We say that a is an 𝒜∗ sequence if for each f∈𝒜 we set AN(f,x)=(1/N)∑i=1Nf({aix})(N=1,2,…), then limN→∞ AN(f,x)=∫01f(t)dt, almost everywhere with respect to Lebesgue measure. Let Vq(f,x)=(∑N=1∞|AN+1(f,x)−AN(f,x)|q)1/q(q≥1). In this paper, we show that if a is an (Lp)∗ for p>1, then there exists Dq>0 such that if ‖f‖p denotes (∫01|f(x)|pdx)1/p, ‖Vq(f,·)‖q≤Dq‖f‖p(q>1). We also show that for any (L1)∗ sequence a and any nonconstant integrable function f on the interval [0,1), V1(f,x)=∞, almost everywhere with respect to Lebesgue measure.