A sequence of independent identically distributed random variables taking values from the set $\{1,2,\dots,N\}$ are partitioned into disjoint intervals of length $n$, and $s$ sequential intervals beginning with the $t$th interval form the $t$th sample of size $ns$. It is proved that if $n\to\infty$ and $N$, $r$ are fixed, then the joint $r$-dimensional distribution of $\chi^2$-statistics constructed for samples of sizes $ns$ with numbers $t_1$ converges to some limit distribution. For this limit distribution, a Gaussian approximation is given. The work was supported by the Russian Foundation for Basic Research, grant 00–15–96136.