Continuous mapping theorem
Encyclopedia
In probability theory
, the continuous mapping theorem states that continuous functions are limit-preserving even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if xn → x then g(xn) → g(x). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {xn} with a sequence of random variables {Xn}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables
.
This theorem was first proved by , and it is therefore sometimes called the Mann–Wald theorem.
s defined on a metric space
S. Suppose a function (where S′ is another metric space) has the set of discontinuity points Dg such that . Then
is equivalent to
Fix an arbitrary closed set F⊂S′. Denote by g−1(F) the pre-image of F under the mapping g: the set of all points x∈S such that g(x)∈F. Consider a sequence {xk} such that g(xk)∈F and xk→x. Then this sequence lies in g−1(F), and its limit point x belongs to the closure
of this set, g−1(F) (by definition of the closure). The point x may be either:
Thus the following relationship holds:
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the continuous mapping theorem states that continuous functions are limit-preserving even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if xn → x then g(xn) → g(x). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {xn} with a sequence of random variables {Xn}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables
Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...
.
This theorem was first proved by , and it is therefore sometimes called the Mann–Wald theorem.
Statement
Let {Xn}, X be random elementRandom element
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line...
s defined on a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
S. Suppose a function (where S′ is another metric space) has the set of discontinuity points Dg such that . Then
Proof
Spaces S and S′ are equipped with certain metrics. For simplicity we will denote both of these metrics using the |x−y| notation, even though the metrics may be arbitrary and not necessarily Euclidian.Convergence in distribution
We will need a particular statement from the portmanteau theorem: that convergence in distribution is equivalent to
Fix an arbitrary closed set F⊂S′. Denote by g−1(F) the pre-image of F under the mapping g: the set of all points x∈S such that g(x)∈F. Consider a sequence {xk} such that g(xk)∈F and xk→x. Then this sequence lies in g−1(F), and its limit point x belongs to the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of this set, g−1(F) (by definition of the closure). The point x may be either:
- a continuity point of g, in which case g(xk)→g(x), and hence g(x)∈F because F is a closed set, and therefore in this case x belongs to the pre-image of F, or
- a discontinuity point of g, so that x∈Dg.
Thus the following relationship holds:
-
Consider the event {g(Xn)∈F}. The probability of this event can be estimated as-
and by the portmanteau theorem the limsup of the last expression is less than or equal to Pr(X∈g−1(F)). Using the formula we derived in the previous paragraph, this can be written as-
On plugging this back into the original expression, it can be seen that-
which, by the portmanteau theorem, implies that g(Xn) converges to g(X) in distribution.
Convergence in probability
Fix an arbitrary ε>0. Then for any δ>0 consider the set Bδ defined as-
This is the set of continuity points x of the function g(·) for which it is possible to find, within the δ-neighborhood of x, a point which maps outside the ε-neighborhood of g(x). By definition of continuity, this set shrinks as δ goes to zero, so that limδ→0Bδ = ∅.
Now suppose that |g(X) − g(Xn)| > ε. This implies that at least one of the following is true: either |X−Xn|≥δ, or X∈Dg, or X∈Bδ. In terms of probabilities this can be written as-
On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {Xn}. The second term converges to zero as δ → 0, since the set Bδ shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem. Therefore the conclusion is that-
which means that g(Xn) converges to g(X) in probability.
Convergence almost surely
By definition of the continuity of the function g(·),-
at each point X(ω) where g(·) is continuous. Therefore-
By definition, we conclude that g(Xn) converges to g(X) almost surely.
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