Consider a succession of random variables where, for each n, $Z_{n}$ is a discrete uniform over $\{1,2, \dots, n\}$. That for each $1\leq k \leq n $, let:
$$X_{n,k}=n^2 1_{\{Z_n=k\}}$$
Finally considerer the succession:
$$\{X_{\{1,1\}},X_{\{2,1\}},X_{\{2,2\}}...,X_{\{n,1\}},...,X_{\{n,n\}}\}$$
Determine in which sense converge to $0$ this succession?
I already know that by Egorovs theorem the succession $f_n(x)=1_{[n,n+1]}(x)$ convergences pointwise to $0$, but i wanna learn if the same argument is valid here.
The sequences converges in probability to zero, since the probability that it differs from zero is decreasing like $1/n$ as each new "row" of the array of variables is reached. The sequence does not converge in expectation to zero, since the expectation of each variable in each "row" is $n$. The sequence does not converge to zero almost surely, since each row of the array of variables takes a nonzero value for each sample point $\omega$ of the underlying sample space.