Convergence in probability of a root of an equation

Question

Let $X_1, \dots, X_n$ be iid Uniform $(0, 1)$ random variables, and set $\theta_n$ to be the root of the equation $$ \sum_{k=1}^n \theta^{X_k} = \sum_{k=1}^n X_k^2. $$ Apparently, this $\theta_n$ converges in probability to a constant. Is that easy to see?

Answer 1

First, we note that the equation $$ \mathsf{E}\theta^X-\mathsf{E}X^2=0,\tag{1}\label{1} $$ where $X\sim U[0,1]$, has the solution $\theta_0=\exp(-W(-3/e^3)-3)$ (here $W$ denotes the the Lambert W function). Replacing expectations on the LHS of Equation $\eqref{1}$ with sample averages yields its finite sample version: $$ \Psi_n(\theta):=\frac{1}{n}\sum_{i=1}^n \left(\theta^{X_i}-X_i^2\right). $$ One may refer to $\hat{\theta}_n$ that solves $\Psi_n(\theta)\approx 0$ as an estimator of $\theta_0$. To show consistency (i.e. $\hat{\theta}_n\xrightarrow{p}\theta_0$) one needs to use the ULLN. However, since $\theta\mapsto\Psi_n(\theta)$ is continuous and nondecreasing, we may resort to Lemma 5.10 in van der Vaart's "Asymptotic Statistics":