Say $X$ is a chi-squared random variable with N degrees of freedom. We know that $X/N \rightarrow 1$ in probability when $N \rightarrow \infty$, due to the law of large numbers.
Now define $y=log(X/M)$. (I don’t know if this matters but note that pdf($y$) has mass at negative values.) Can it be shown that $y\rightarrow 0$ (for example in probability) as $N \rightarrow \infty$?
Remark: I guess the "right" way is to find the pdf($y$) by the transformation theorem, and use moment generation functions to show convergence. But is there any simpler way to prove it?
I assume you mean $Y = \log(X/N)$? Then $Y \xrightarrow{P} 0$ holds indeed and it is a simple consequence of the Continuous mapping theorem.