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Consider one $1-$sub-gaussian or Bernoulli variables $X$, we i.i.d. sample $X$ $n$ times $(X_1,...,X_n)$; and $\mu_X<S$, $S$ is a constant threshold. We have such a quantity: $N_X = \sum_{i=1}^n \mathbb{I} (\hat{\mu}_X(i) < S)$, where the empirical mean is $\hat{\mu}_X(i) = \frac{\sum_{j=1}^iX_j}{i}$. The frequency $N_X$ describes how many times the empirical average of $X$ is below such threshold $S$ during the sampling process.

Here, we want to calculate or bound the probability of the frequency equal to or smaller than some constant $\ell$, i.e., $P(N_X = \ell)$ or $P(N_X<\ell)$ using $\mu_X, S, n$ these quantities. Since the added indicators are dependent, it is difficult to compute the probability distribution directly.

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  • $\begingroup$ you take as given that the mean of $X$ is smaller than $S$, i.e. that $\mu_{X}<S$? $\endgroup$
    – Fiodor1234
    Commented Jul 31 at 13:22
  • $\begingroup$ Yes, we do. As the problem formulation. $\endgroup$
    – white
    Commented Jul 31 at 17:18
  • $\begingroup$ And we can consider whatever 1-subgaussian we want? For example a uniform on $U[-1,1]$? $\endgroup$
    – Fiodor1234
    Commented Jul 31 at 17:49

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