I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ distributed.} } \right) \right] $$ where ${\cal N}$ is a n-dimensional multivariate normal distribution, $[{\cal N}]_1$ the first coordinate of ${\cal N}$ and $\ln^-( x ) = | \ln(x) 1_{\{x \leq 1\}}| $. I know that almost surely $\lim_n n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \| {\cal N} \|^2 \right) = (2 \sigma [ {\cal N}]_1 - \sigma^2) 1_{\{ 2 [ {\cal N}]_1 - \sigma \geq 0 \}}$ (where I use $\frac{1}{n}\| {\cal N}\|^2$ goes to $1$ from the Law of Large Numbers). I however did not succeed to prove the uniform integrability of the sequence $ Y_n = n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \| {\cal N} \|^2 \right)$ . I tried in particular to use the integral expression of the expectation (that I obtained using spherical coordinates): $$ n F\left(\frac{\sigma}{n} \right) = \frac{1}{4} \frac{n}{W_{n-2}} \int_0^{\frac{\pi}{2}} \!\! \int_0^{+\infty} \ln^{-}\left(1 - 2 \frac{\sigma}{n} \sqrt{r} \cos(\theta) + \frac{\sigma^2}{n^2} r\right) \sin^{n-2}(\theta) \underbrace{ \frac{\exp(-\frac{r}{2}) r^{\frac{n}{2}-1} }{\Gamma(\frac{n}{2}) 2^{n/2}}}_{\mbox{density of } \chi^2} d \theta dr $$ where $W_{n-2}= \int \sin^{n-2}(\theta)$ is the Wallis integral but it didn't help me. One problem I have, is that if I try to control the error of the Taylor approximation of $\ln^-(1+x)$ by $-x$ in the integral, I have a singularity that is not integrable.
Any hints, suggestions would be welcome!