I come up with the question while self-studying. The following is from Keener(2010) Theoratical Statistics.
$X_i \overset{iid}{\sim} \text{Uniform} (0,1) \ (i=1,\cdots,n)$
Let $T_n := \arg\max_{t>0} \sum_{i=1}^{n} \frac{\log(1+t^2 X_i)}{t}$
Then the question is to find out $T_n \overset{p}{\rightarrow}c\in \mathbb{R}^n$ and the asymptotic distribution for $\sqrt{n}(T_n-c)$.
In order to obtain the argmax, $\frac{\partial}{\partial t} \mid_{t=T_n} \sum_{i=1}^{n} \frac{\log(1+t^2 X_i)}{t} = \sum_{i=1}^{n} \left[ \frac{2 X_i}{1+T_{n}^2 X_i} - \frac{\log (1+T_{n}^2 X_i)}{T_{n}^2} \right] = 0$. But the resulting form is rather hideous.
On the other hand, I thought of deriving the pdf of $\sum_{i=1}^{n} Y_i := \sum_{i=1}^{n} \frac{\log(1+t^2 X_i)}{t}$ since $Y_n$ are iid with pdf $f_Y (y) = \frac{e^{yt}}{t} I_{\left[0, \frac{\log(1+t^2)}{t}\right]} (y)$. I may think of using convolution, but not sure of I'm on the right track.
Any hint would welcome.