# Probability of large deviations

2010 Mathematics Subject Classification: Primary: 60F10 [MSN][ZBL]

A probability of the type

$${\mathsf P} ( S _ {n} - b _ {n} > a _ {n} ),\ \ {\mathsf P} ( S _ {n} - b _ {n} < - a _ {n} ) \ \ \textrm{ or } \ {\mathsf P} (| S _ {n} - b _ {n} | > a _ {n} ),$$

where

$$S _ {n} = \sum _ { j= } 1 ^ { n } X _ {j} ,$$

$\{ X _ {j} \}$ is a sequence of independent random variables, and $\{ a _ {n} \}$ and $\{ b _ {n} \}$ are two numerical sequences such that $a _ {n} > 0$, and $( S _ {n} - b _ {n} ) / a _ {n} \rightarrow 0$ in probability.

If the random variables $X _ {1} , X _ {2} \dots$ have the same distribution with mathematical expectation zero and finite variance $\sigma ^ {2}$, one may write $b _ {n} = 0$ and $a _ {n} = x _ {n} \sigma \sqrt n$, where $x _ {n} \rightarrow + \infty$ as $n \rightarrow \infty$. Cramér's theorem and strengthened versions of it are particularly important in this connection (cf. Cramér theorem).

To obtain guaranteed bounds for the probability of large deviations one uses inequalities of the type of the Chebyshev inequality in probability theory; these provide the so-called exponential bounds for the probability of large deviations. For instance, if the random variables $X _ {j}$ are independent, ${\mathsf E} {X _ {j} } = 0$, ${\mathsf E} {X _ {j} ^ {2} } = \sigma _ {j} ^ {2}$, $| X _ {j} | \leq L$ with probability one, $B _ {n} ^ {2} = \sigma _ {1} ^ {2} + \dots + \sigma _ {n} ^ {2}$ and $a = xL/B _ {n}$, then the estimate

$${\mathsf P} \{ | S _ {n} | > x B _ {n} \} \leq \ 2 \mathop{\rm exp} \left \{ - \frac{x ^ {2} }{2} \left ( 1+ \frac{a}{3} \right ) ^ {-} 1 \right \} ,$$

the right-hand side of which decreases exponentially with increasing $x$, is valid for all $x \geq 0$.

#### References

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