Abstract. We derive a simple formula expressing the expected maximal interval between rare random events in terms of the average interval:
where a is the average interval between the rare events, and T is the total observation time or length, as applicable.
E(max interval) = a ln(T/a) + O(a) (1 << a << T),
The formula holds for random events occurring at
exponentially distributed (real-valued) intervals, as well as
for events at
geometrically distributed discrete (integer-valued) intervals.
(For more information on extreme value distributions of random sequences see ;
for extreme value distributions of integer-valued random sequences,
such as head runs in coin toss sequences, see [5,6] and further references therein.)
We expect that similar formulas would also be applicable to maximal intervals between terms
in certain other kinds of random or pseudorandom sequences.
Moreover, a similar heuristic formula
Notation. In all formulas hereafter:
ln x denotes the natural logarithm of x
logbx denotes the base-b logarithm of x
E(x) denotes the expected value of a random variable x
SD x denotes the standard deviation of x
Var x denotes the variance of x
y = O(x) is shorthand for y is such that |y| ≤ Cx for some constant C.
y = o(x) is shorthand for y is such that y/x tends to zero.
Problem A. Consider a non-stop toll bridge with very light traffic.
Problem B. Consider a biased coin with a probability of heads
In both problems, we want to answer the following questions:
(1) What is the expected total number of rare events (cars or tails) in the observation series of length T?
(2) What is the expected average interval a between events (i.e. between cars/tails)?
(3) What is the expected maximal interval between events, as a function of the average interval a?
Before you read on, try to answer the questions yourself. (Notice that the first and the second question are so much easier than the third!)
Here are the easy answers:
(1) Because the probability of the event is q at any given second/toss, we expect a total of nq events after n seconds/tosses, and a total of Tq events at the end of the entire observation series of length T.
(2) To estimate the expected average interval a between events, we divide the total
length T of our observation series by the expected total number of events Tq.
So the expected average interval between events is
(3) Quite obviously, we can predict that the expected maximal interval is less than T, but not less than a:
a ≤ E(max interval) < T.The expected maximal interval will likely depend on both a and T:
E(max interval) = ƒ(a,T).It is also reasonable to expect that ƒ(a,T) is an increasing function of both arguments, a and T.
This is nice but can we say anything more specific about the expected maximal interval?
In problem A, to estimate the most probable maximal interval between cars we proceed as follows:
After n seconds of observations, we would have seen about nq cars, hence about nq intervals between cars. The intervals are independent of each other and real-valued. A known good model for the distribution of these intervals is the exponential distribution:
with probability 1, each interval must be at least 0 seconds;
with probability p, any given interval is at least 1 second;1
with probability p2, any given interval is at least 2 seconds;
with probability p3, any given interval is at least 3 seconds;
with probability pt, any given interval is at least t seconds.
Thus, after n seconds of observations and about nq carless intervals,
we would reasonably expect that at least one interval is
no shorter than
Now it is easy to estimate the most probable maximal interval tmax:
pt × (nq) ≥ 1.
ptmax ≈ 1/(nq)
(1/p)tmax ≈ nq
tmax ≈ log1/p(nq).
In problem B we can estimate the longest run of heads Rn after n coin tosses reasoning very similarly. One notable difference is that now the head runs are discrete (have integer lengths). Accordingly, they should be modeled using the geometric distribution. Schilling  gives this estimate for the longest run of heads after n tosses, given the heads probability p:
In both problems, the estimates for the most probable maximal interval (as a function of p and n) have the same form
Rn ≈ log1/p(nq).
or, omitting the o(q) terms,
ln(1/p) = ln(1/(1 − q)) = ln(1 + q + o(q)) = q + o(q)
ln(1/p) ≈ ln(1 + q) ≈ q, and therefore
1 / ln(1/p) ≈ 1/q ≈ a.
This allows us to transform our previous estimate log1/p(nq) as follows:
For a long series of observations, with the total length or duration
log1/p(nq) = ln(nq) / ln(1/p) ≈ (1/q) ln(nq) ≈ a ln(n/a).
(A) Exponential initial distribution.
Fisher and Tippett , Gnedenko , Gumbel  and
showed that, for initial distributions of exponential type (including the
exponential distribution as a special case)
the limiting distribution of maximal terms in a random sequence is
the double exponential distribution
often called the Gumbel distribution.
If the initial exponential distribution has the mean a and the probability density function (PDF)
where γ = 0.5772... is the Euler-Mascheroni constant. The mean value of observed maximal intervals in problem A will converge almost surely to the mean
Scale = a = 1/ln(1/p) ≈ 1/q
Mode = μ = log1/p(Tq) ≈ a ln(T/a)
Mean = μ + γa
E(max interval) = μ + γa = log1/p(Tq) + γ/ln(1/p) or, expressing the result in terms of a and T,
E(max interval) ≈ a ln(T/a) + γa = a ln(T/a) + O(a).
Interestingly, Gumbel was not the original discoverer of his namesake distribution.
Fisher and Tippett (1928) 
described three types of limiting extreme-value distributions and
showed that the double exponential distribution
is the limiting extreme-value distribution for a certain wide class of random sequences.
They also computed, among other parameters, the mean-to-mode distance in the
double exponential distribution (see , p.186; it is this result that allows one to conclude that the mean is
(B) Geometric initial distribution. Somewhat surprisingly, in this case the limiting extreme-value distribution does not exist (see , p.203, for a detailed explanation). For the longest run of heads Rn in a series of n tosses of a biased coin, with the probability of heads p, Schilling  gives the expected value
where the first term is the same as in Problem A (up to a substitution
E(Rn) = log1/p(nq) + γ/ln(1/p) − 1/2 + r1(n) + ε1(n)
E(Rn) = a ln(n/a) + O(a).
(A) Exponential initial distribution.
Here the limiting distribution of maximal intervals is
the Gumbel distribution
with the scale
(B) Geometric initial distribution. For the longest run of heads Rn in a series of n tosses of a biased coin, the variance is
where the first term grows as
Var Rn = π2/(6 ln2(1/p)) + 1/12 + r2(n) + ε2(n) (, p.202)
SD Rn = π/(√ln(1/p)) + O(1) = O(a).
These two situations are somewhat different:
in case (A) maximal intervals have a limiting distribution (the Gumbel distribution), while
in case (B) no limiting distribution exists (here the Gumbel distribution is simply a good approximation).
Nevertheless, in both cases the expected maximal interval between events is
where a is the average interval between events, and T is the total observation time or length. When
E(max interval) = a ln(T/a) + γa + lower-order term = a ln(T/a) + O(a), with standard deviation ≈ πa/√= O(a),
A remarkably similar heuristic formula,
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Proc. Camb. Phil. Soc., 24 (1928), 180-190.
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Ann. Math., 44 (1943), 423-453.
English translation: On the limiting distribution of the maximum term in a random series. In: Breakthroughs in Statistics, Volume 1: Foundations and Basic Theory. Springer, New York (1993), 185-225.
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4. E.J. Gumbel, Statistics of Extremes, Columbia University Press, New York (1958)
5. M. F. Schilling, The longest run of heads. The College Mathematics Journal, v.21 (1990), No.3, 196-207.
6. L. Gordon, M.F. Schilling, and M.S. Waterman, An extreme value theory for long head runs. Probability Theory and Related Fields, v.72 (1986), 279-297.
8. A. Kourbatov, Maximal gaps between Cramer's random primes (2014)