Suppose $X_1,\cdots,X_n,\cdots$ are i.i.d. and follow the uniform distribution on $(0,1)$: $U(0,1)$. Define $T_n$ to be the maximum point of the function $$f_n(t)=\sum_{i=1}^{n}\frac{\log(1+t^2X_i)}{t}.$$
(1) Show that $T_n$ converges to a constant $c$ in probability.
(2) Find the limiting distribution of $\sqrt{n}(T_n-c)$.
I encountered this problem when I was reviewing for the exam of statistical inference, and this problem appeared in the exam in 2022, but I didn't see how statistical inference is related to this.
Anyway, I discussed this problem with my classmates and only thing we did is to write $T_n$ as the solution of the equation $$0=t^2f_n'(t)=\sum_{i=1}^{n}\left(\frac{2t^2X_i}{1+t^2X_i}-\log(1+t^2X_i)\right).$$ Then we found it pretty hard to proceed. Indeed we didn't see how to use the standard results (e.g. Laws of large numbers or Central limit theorems) or how to apply to the standard trick(e.g. calculate the characteristic function or the generating function of $T_n$).
Any help will be appreciated.
$T_n$ is an M-estimator, as it is the argmax of the following random function:
$$f_n(t)=\sum_{i=1}^{n}\frac{\log(1+t^2X_i)}{t}.$$
Note that MLEs are also M-estimators.
Under some regularity conditions, an M-estimator $M_n$ is consistent, that is, $M_n$ converges to some point $c$ in probability, and a normalized version of $M_n$ is asymptotically normally distributed, that is, $\sqrt{n}(M_n-c)$ converges to a normal distribution in law. See here for more details.
The following properties of the function:
$$h(t)=\frac{\log(1+at^2)}{t}$$
with $a\in [0,1]$, help you to show that all the regularity conditions are satisfied:
1- $h(t)$ is bounded by a constant (the maximum value of $h(t)$ for $a=1$).
2- $h(t)$ is Lipchitz continuous with $K=2$, as
$$-a \le h'(t) \le 2a \rightarrow |h'(t)| \le 2a \le 2.$$
3- $h(t)$ has a unique maximizer, because $h(0^+)=0, h(\infty)=1$, it is continuous, and its derivative is a continuous function taking first positive and then negative values (it becomes zero at a single point).
These can be used to prove the same properties for $f_n(t)$.