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Let $h$ be some bounded non-negative function. Assume that some random quantity $\mu^N (h)$ be some random quantity with almost sure limit $\mu(h) > 0$. For instance we could have $\mu^N(h) = N^{-1} \sum_{i = 1}^N h(X^i)$ with $X^{1:N} \overset{i.i.d.}{\sim} \mu$.

Furthermore, assume that the difference $\mu^N(h) - \mu(h)$ is subgaussian, i.e. $$ \mathbb{P}( \big| \mu^N(h) - \mu(h) \big| > \epsilon ) \leq \exp(-CN\epsilon ^2) .$$   As a consequence, the $L^2$ norm of the difference goes to zero. I am wondering if this also the case for the inverse, i.e. does $ \left\| \frac{1}{\mu^N(h)} - \frac{1}{\mu(h)} \right\|_2$ go to zero when N goes to infinity?

If $h$ is lower bounded this is true, but i would like to know if this is also the case without this assumption.

I've tried to conjecture the result for the sample mean and it seems that the $L2$ norm goes to zero as long as the limit $\mu(h)$ is not too small.

yprobnoob
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  • Your notation is ambiguous. In particular, is $\mu(h)$ a number or a random variable or neither? – whuber Jan 29 '22 at 15:38
  • @whuber sorry for the ambiguity. $\mu(h)$ is a real number. – yprobnoob Jan 29 '22 at 15:47
  • What are you assuming about the relationship between $\mu^N(h)$ and $h$? If nothing, then why even mention $h$ at all? What is the point of assuming $h$ is bounded? It would seem you are trying to ask about certain kinds of sequences of random variables $X_N,$ $N=1,2,\ldots,$ but the notation suggests you might have something else in mind. – whuber Jan 29 '22 at 16:18
  • i mentioned $h$ so as to give an example of what i'm talking about (the example of the sample mean). I also assumed $h$ to be upper bounded because it is a sufficient condition to have a Hoeffding inequality for the sample mean. I indeed have something else in mind but i'd rather figure out how to make it work for the sample mean and then work out the remaining details for my particular case by myself. – yprobnoob Feb 01 '22 at 12:54
  • Since you're trying to simplify your actual problem, then it would be better to remove the extraneous stuff: it only complicates what might be a simple situation. – whuber Feb 01 '22 at 14:23

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