Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that:
$$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$
My question is, what type of probabilistic inequalities require some limit on the sample size such that the inequalities apply? i.e. something similar to:
$$m>f(\delta) \implies \text{some probabilistic inequality holds}$$
Context:
I am trying to do some question and trying to apply Chebyshev's inequality or the law of large numbers (LLN). I am given that the sample size m should be greater than some bound that they give in terms of an error term that I can choose and some other terms that are not important for this question. My confusion is that the different inequalities I have tried look something like:
$$Pr[|M_m-\mu| \geq \epsilon] \leq \frac{\sigma^2}{m \epsilon^2}$$
and it doesn't really require that m must be greater than some value, so I had a suspicion that maybe there is something about Chebyshev's that I don't know or something about LLN that I don't know.
I am looking for a different list of inequalities that apply given some bound on the sample size or something like that. Or a list of the different forms of the LLN and Chebyshev's inequality.
So far I think the only bounds I have found that have explicit conditions on the sample size is something called "convergence" probability i.e.:
For a sequence of r.v. $Y_m$ for every accuracy level $\epsilon$ and confidence level $\delta$ and $m \geq m_0$ (i.e. sufficiently large m) we have:
$$Pr[|Y_m - a|\geq \epsilon] \leq \delta$$
Though I know little about this bound, so if anyone knows its relation to LLN or Chebyshev's it would be awesome! Also, I was unsure how you even relate $m_0$ to the confidence level, how is that manipulation done?
If you are interested in seeing what the actual/original question is, I will provide a link to it:
How to use Chebyshev's inequality or the law of large numbers to a probability question
Well, if I understood you correctly, I think Markov's inequality and Chernoff bound in addition to Chebyshev's inequality with the strong and weak laws of large numbers inequality can help you in the beginning. :)