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The Hoeffding inequality turns finite interval bounds and independence into exponentially small tail probabilities for a sum. It requires neither a normal distribution nor identical probability distributions. Let $X_1,\dots ,X_n$ be independent essentially bounded real random variables, with deterministic finite bounds $a_i\le X_i\le b_i$ almost surely. Define the centered sum and its squared-width budget byFor $V>0$ and $t>0$,The union bound therefore gives $\mathbb{P}(|S_n|\ge t)\le \min \{1,2e^{-2t^2/V}\}$. If $V=0$, every $X_i$ is constant almost surely, so $S_n=0$ and both positive-threshold tail probabilities vanish. The equal-width version is stated in the authored Stanford notes.^[ref]

The quantity $V$ is a sum of squared interval widths, not the variance of $S_n$. Centering changes the endpoints to $a_i-\mathbb{E}{X}_i$ and $b_i-\mathbb{E}{X}_i$, leaving each width unchanged. For example, independent random variables in $[-2,1]$ and $[4,5]$ have $V=3^2+1^2=10$, whatever their expected values or probability distributions. A deviation at least three above their total expected value has probability at most $e^{-18/10}\approx 0.165299$. If all widths equal $b-a>0$, the sample mean ${\overline{X}}_n=n^{-1}{\sum }_i{X}_i$ satisfies

Independence is essential to this general guarantee. Take one Bernoulli random variable $B$ with success probability $1/2$ and set every $X_i=B$. Each lies in $[0,1]$, but for every $n$ the sample mean is either zero or one. Thus $\mathbb{P}(|{\overline{X}}_n-1/2|\ge 1/4)=1$, whereas incorrectly inserting these dependent observations into the formula would give $2e^{-n/8}$, less than one for $n\ge 6$. Some special dependent models have their own valid concentration inequalities; arbitrary statistical dependence cannot be ignored.

For the short proof, apply Markov's inequality to $e^{\lambda S_n}$ with $\lambda >0$, factor the expectation using independence, and apply the Hoeffding lemma to each factor:Minimizing this quadratic polynomial at $\lambda =4t/V$ gives the upper tail probability; applying it to $-S_n$ gives the lower one. This is the Chernoff bound method. The original 1963 article is Probability Inequalities for Sums of Bounded Random Variables by Wassily Hoeffding; its publisher record identifies the article, but its full text was unavailable during this source check. The argument above is a direct derivation rather than a claim to have inspected that article's proof.

Author of probability inequalities for sums of bounded independent random variables. The resulting Hoeffding inequality bounds tail probabilities without requiring an exact distribution for the sum, making it useful in concentration of measure.^[ref]