  Let $f$ be a locally integrable function defined on $\mathbb{R}$, and let
$(n_k)$ be a lacunary sequence. Define the operator $A_{n_k}$ by
$$A_{n_k}f(x)=\frac{1}{n_k}\int_0^{n_k}f(x-t)\, dt.$$ We prove various types of
new inequalities for the variation operator
$$\mathcal{V}_sf(x)=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^s\right)^{1/s}$$
when $2\leq s<\infty$.
