# Probabilistic Bounds on the Frequency of Empirical Mean Threshold Crossings for Sub-Gaussian or Bernoulli Variables

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$$ is a constant threshold. We have such a quantity: $$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.

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