In probability theory, a property is said to hold almost surely if it holds for all sample points, except possibly for some sample points forming a subset of a zero-probability event.
If an event happens almost surely, then it is called an almost sure event.
The abbreviations a.s. (almost sure or almost surely) and w.p. 1 (with probability 1) are used very frequently.
The concept is defined as follows.
Definition
Let
be a sample space. Let
be the set of its points satisfying a given
property:
Property
is said to be an almost sure property if the set
of all points that do not satisfy property
is included in a zero-probability event
,
i.e.,
Remember that not all subsets of the sample space are necessarily considered
events (see the lecture entitled Probability).
Therefore, the set
needs not necessarily be an event. In case it is, it is called an almost sure
event.
Suppose the sample space is the unit interval, that
is,and
suppose that the
set
is
a zero-probability
event:
Now
consider the
function
The
function is well-defined on the interior of
but not at its endpoints
(
and
),
where it tends to infinity. Can we say that
is almost surely well-defined on
?
Define the
set
Then
the complement of
is
Clearly,
we have
that
and
we know that
is a zero-probability event. As a consequence, we can say that
is almost surely well-defined on
.
A more articulated explanation, as well as an extensive discussion of the subtleties of the concept of zero-probability event, is provided in the lecture entitled Zero-probability events.
Previous entry: Adjusted sample variance
Next entry: Alternative hypothesis
Please cite as:
Taboga, Marco (2021). "Almost sure", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/almost-sure.
Most of the learning materials found on this website are now available in a traditional textbook format.