Bounds on the Expectation of the Maximum of
Samples from a Gaussian
Gautam Kamath
In this document, we will provide bounds on the expected maximum of n
samples from Gaussian distribution. Let Y = max1≤i≤n Xi , where Xi are iid
random variables, distributed as N (0, σ 2 ). We will show that
p
√ p
1
log n ≤ E[Y ] ≤ σ 2 log n
π log 2
√ √
First, we show E[Y ] ≤ σ 2 log n.
σ√
exp{tE[Z]} ≤ E[exp{tZ}]
= E[max exp{tXi }]
n
X
≤
E[exp{tXi }]
i=1
= n exp{t2 σ 2 /2}
The first inequality is Jensen’s inequality, the second is the union bound, and
the final equality follows from the definition of the moment generating function.
Taking the logarithm of both sides of this inequality, we get
E[Z] ≤
log n tσ 2
+
t
2
√
This can be minimized by setting t =
2 log n
,
σ
which gives us the desired result,
√ p
E[Z] ≤ σ 2 log n
√
Next, we show the more difficult direction - that E[Y ] ≥ σC log n, where
C = √π 1log 2 . If one of our n samples is greater than some value, then the
maximum is at least that value. Thus, we lower bound the expectation
√ of Y by
showing that we expect at least one of the Xi to be greater than σC log n.
1
Let k =
2C 2
π .
E[|{i : Xi ≥ σC
p
p
log n}|] = nP (Xi ≥ σC log n)
p
n
= P (|Xi | ≥ σC log n)
2
n
C p
=
1 − erf √
log n
2
2
p
n
≥
1 − 1 − exp (−k log n)
2
p
n
1 − 1 − n−k
=
2
Where the second equality follows by symmetry, the third equality is based on
the CDF of the half-normal distribution,
and the inequality is from the bound
q
on the error function, erf (x) ≤ 1 − exp − π4 x2 . We require this value to be
at least 1:
p
n
⇔
1 − 1 − n−k ≥ 1
2
p
2
1 − ≥ 1 − n−k
⇔
n
4
4
1
1− + 2 ≥1− k
⇔
n n
n
n2−k ≥ 4n − 4
⇔
(2 − k) log n ≥ log(4n − 4)
⇔
log(4n − 4)
2−
≥k
log n
Substituting in the value C = √π 1log 2 tell us this inequality holds for n ≥ 5.27.
Since we are only concerned with integer values of n, this inequality holds
√ for
n ≥ 6. We can confirm the inequality for 1 ≤ n ≤ 5 by comparing σC log n
with the explicit equations for E[Y ]. √
Note that this inequality is tight for n = 2.
Therefore, we have E[Y ] ≥ σ √π 1log 2 log n.
2