Etemadi's inequality

In probability theory, Etemadi's inequality is a so-called "maximal inequality", an inequality that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. The result is due to Nasrollah Etemadi.

Statement of the inequality

Let X1, ..., Xn be independent real-valued random variables defined on some common probability space, and let α ≥ 0. Let Sk denote the partial sum

S

k

=

X

1

+

⋯

+

X

k

.

{\displaystyle S_{k}=X_{1}+\cdots +X_{k}.\,}

Then

Pr

(

max

1

≤

k

≤

n

|

S

k

|

≥

3

α

)

≤

3

max

1

≤

k

≤

n

Pr

(

|

S

k

|

≥

α

)

.

{\displaystyle \Pr {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq 3\alpha {\Bigr )}\leq 3\max _{1\leq k\leq n}\Pr {\bigl (}|S_{k}|\geq \alpha {\bigr )}.}

Remark

Suppose that the random variables Xk have common expected value zero. Apply Chebyshev's inequality to the right-hand side of Etemadi's inequality and replace α by α / 3. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side:

Pr

(

max

1

≤

k

≤

n

|

S

k

|

≥

α

)

≤

27

α

2

var

⁡

(

S

n

)

.

{\displaystyle \Pr {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq \alpha {\Bigr )}\leq {\frac {27}{\alpha ^{2}}}\operatorname {var} (S_{n}).}

References

Billingsley, Patrick (1995). Probability and Measure.