Let's denote a chi-squared distribution with $k$ degrees of freedom as $\chi^2_k$, and a random variable $Z_k=\frac{X_k-k}{\sqrt{2k}}$, where $X_k\sim\chi^2_k$. Thus, $Z_k$ is a chi-squared random variable that is normalized to zero mean and unity variance.

It's well known that a chi-squared random variable with $k$ degrees of freedom can be expressed as a sum of $k$ i.i.d. chi-squared random variables each with one degree of freedom, i.e. $X_k=\sum_{i=1}^k Y_i$ with $Y_i\sim\chi^2_1$. The fact that chi-squared random variable with one degree of freedom has bounded moment generating function around zero implies that the right tail of $Z_k$ is normal even in the large deviation regime, when $x=o(\sqrt{k})$: $P\left(Z_k>x\right)\approx1-\Phi(x)$ where $\Phi(x)=\int_0^x\frac{e^{-t^2/2}}{\sqrt{2\pi}}dt$ is the standard normal cumulative density function. This is per Chapter 8 of Petrov's "Sums of Independent Random Variables" (1975).

Also, since the probability density function of $Z_k$, $f_{Z_k}(x)$, is bounded for $k>2$, per my understanding of Theorem 7 in Chapter 7 of the aforementioned book by Petrov, a local limit theorem holds for $Z_k$, whereby $\sup_x|f_{Z_k}(x)-\phi(x)|\rightarrow 0$ with $\phi(x)=\frac{e^{-x^2/2}}{\sqrt{2\pi}}$ being the standard normal density function.

I am wondering if a local limit theorem applies to the exponent of $Z_k$, i.e. is there a result that the density $f_{\exp[Z_k]}(x)$ of $e^{Z_k}$ point-wise converges to that of a standard log-normal random variable $f_{\log\mathcal{N}}(x)=\frac{e^{-\frac{\log^2x}{2}}}{x\sqrt{2\pi}}$? Formally, is the following true: $\sup_x|f_{\exp[Z_k]}(x)-f_{\log\mathcal{N}}(x)|\rightarrow 0$? I admit that I am very new to local limit theorems and don't know the area very well...


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