  This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set
$X\subset \mathbb{R}^d$, a random variable $\omega_Q\in[0,1]$ is assigned for
each elementary cube $Q$ in $X$, and a random cubical set $X(t)$ is defined by
the sublevel set of $X$ consisting of elementary cubes with $\omega_Q\leq t$
for each $t\in[0,1]$. Under this setting, the main results of this paper show
the limit theorems (law of large numbers and central limit theorem) for Betti
numbers and lifetime sums of random cubical sets and filtrations. In addition
to the limit theorems, the positivity of the limiting Betti numbers is also
shown.
