Q: Plims of matrices - plim X'X/n =Q Let $X$ be a $n \times 2$ matrix with column 1 as a column of all 1's and column 2 a column of IID random variables $Z_1, Z_2,....Z_n$ where $Z_i$ is Bernoulli distributed with parameter $\theta$
What is the 
$$\textrm{p}\lim_{n\to \infty
}  X′X/n$$ ?
Where $\textrm{p}\lim$ is convergence in probability.
The resulting matrix will of course be a $2 \times 2$ matrix with first element as 1, the 2nd element of the first row and 1st element of the second row are $\frac{1}{n}\sum_i Z_i$ and the last element is $\frac{1}{n}\sum_i(Z_i)^2$. What is the $\textrm{p}\lim$ of this matrix? And what value of $n$ will make it singular?
I know that since the $Z$'s are Bernoulli variables, their sum follows a binomial distribution and will have finite second moments. The $\textrm{p}\lim$ will therefore be a positive definite matrix. But I dont know what the matrix will be. Can someone help?

A: Denote the sample mean by $\bar{Z}_n = \frac{1}{n} \sum_{i=1}^n Z_i$.  Expanding out the relevant expression for your design matrix, you can easily show that:
$$\frac{\mathbf{X}' \mathbf{X}}{n} = 
\begin{bmatrix}
1 & \bar{Z}_n \\
\bar{Z}_n & \bar{Z}_n \\ 
\end{bmatrix}.$$
(Note here that $Z_i$ is binary, so you have $\sum Z_i^2 = \sum Z_i$, which allows you to simplify the last element of the matrix to the above form.)
Since $Z_1, Z_2, Z_3, ... \sim \text{IID Bern}(\theta)$ it follows from the weak law of large numbers that $\text{plim}_{n \rightarrow \infty} \bar{Z}_n = \theta$, which means you have:
$$\underset{n \rightarrow \infty}{\text{plim}} \text{ } \frac{\mathbf{X}' \mathbf{X}}{n} = 
\begin{bmatrix}
1 & \theta \\
\theta & \theta \\ 
\end{bmatrix}.$$
