Continuous Mapping of Interval Onto Itself

In probability theory, the continuous mapping theorem states that continuous functions preserve limits 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 Henry Mann and Abraham Wald in 1943,[1] and it is therefore sometimes called the Mann–Wald theorem.[2] Meanwhile, Denis Sargan refers to it as the general transformation theorem.[3]

Statement [edit]

Let {Xn }, X be random elements defined on a metric space S. Suppose a function g: SS′ (where S′ is another metric space) has the set of discontinuity points Dg such that Pr[X ∈Dg ] = 0. Then[4] [5]

X n d X g ( X n ) d g ( X ) ; X n p X g ( X n ) p g ( X ) ; X n a.s. X g ( X n ) a.s. g ( X ) . {\displaystyle {\begin{aligned}X_{n}\ {\xrightarrow {\text{d}}}\ X\quad &\Rightarrow \quad g(X_{n})\ {\xrightarrow {\text{d}}}\ g(X);\\[6pt]X_{n}\ {\xrightarrow {\text{p}}}\ X\quad &\Rightarrow \quad g(X_{n})\ {\xrightarrow {\text{p}}}\ g(X);\\[6pt]X_{n}\ {\xrightarrow {\!\!{\text{a.s.}}\!\!}}\ X\quad &\Rightarrow \quad g(X_{n})\ {\xrightarrow {\!\!{\text{a.s.}}\!\!}}\ g(X).\end{aligned}}}

where the superscripts, "d", "p", and "a.s." denote convergence in distribution, convergence in probability, and almost sure convergence respectively.

Proof [edit]

This proof has been adopted from (van der Vaart 1998, Theorem 2.3) harv error: no target: CITEREFvan_der_Vaart1998 (help)

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 Euclidean.

Convergence in distribution [edit]

We will need a particular statement from the portmanteau theorem: that convergence in distribution X n d X {\displaystyle X_{n}{\xrightarrow {d}}X} is equivalent to

E f ( X n ) E f ( X ) {\displaystyle \mathbb {E} f(X_{n})\to \mathbb {E} f(X)} for every bounded continuous functional f.

So it suffices to prove that E f ( g ( X n ) ) E f ( g ( X ) ) {\displaystyle \mathbb {E} f(g(X_{n}))\to \mathbb {E} f(g(X))} for every bounded continuous functional f. Note that F = f g {\displaystyle F=f\circ g} is itself a bounded continuous functional. And so the claim follows from the statement above.

Convergence in probability [edit]

Fix an arbitrary ε > 0. Then for any δ > 0 consider the set Bδ defined as

B δ = { x S x D g : y S : | x y | < δ , | g ( x ) g ( y ) | > ε } . {\displaystyle B_{\delta }={\big \{}x\in S\mid x\notin D_{g}:\ \exists y\in S:\ |x-y|<\delta ,\,|g(x)-g(y)|>\varepsilon {\big \}}.}

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 lim δ → 0 Bδ  = ∅.

Now suppose that |g(X) −g(Xn )| >ε. This implies that at least one of the following is true: either |XXn | ≥δ, or X ∈Dg , or XBδ . In terms of probabilities this can be written as

Pr ( | g ( X n ) g ( X ) | > ε ) Pr ( | X n X | δ ) + Pr ( X B δ ) + Pr ( X D g ) . {\displaystyle \Pr {\big (}{\big |}g(X_{n})-g(X){\big |}>\varepsilon {\big )}\leq \Pr {\big (}|X_{n}-X|\geq \delta {\big )}+\Pr(X\in B_{\delta })+\Pr(X\in D_{g}).}

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

lim n Pr ( | g ( X n ) g ( X ) | > ε ) = 0 , {\displaystyle \lim _{n\to \infty }\Pr {\big (}{\big |}g(X_{n})-g(X){\big |}>\varepsilon {\big )}=0,}

which means that g(Xn ) converges to g(X) in probability.

Almost sure convergence [edit]

By definition of the continuity of the function g(·),

lim n X n ( ω ) = X ( ω ) lim n g ( X n ( ω ) ) = g ( X ( ω ) ) {\displaystyle \lim _{n\to \infty }X_{n}(\omega )=X(\omega )\quad \Rightarrow \quad \lim _{n\to \infty }g(X_{n}(\omega ))=g(X(\omega ))}

at each point X(ω) where g(·) is continuous. Therefore,

Pr ( lim n g ( X n ) = g ( X ) ) Pr ( lim n g ( X n ) = g ( X ) , X D g ) Pr ( lim n X n = X , X D g ) = 1 , {\displaystyle {\begin{aligned}\Pr \left(\lim _{n\to \infty }g(X_{n})=g(X)\right)&\geq \Pr \left(\lim _{n\to \infty }g(X_{n})=g(X),\ X\notin D_{g}\right)\\&\geq \Pr \left(\lim _{n\to \infty }X_{n}=X,\ X\notin D_{g}\right)=1,\end{aligned}}}

because the intersection of two almost sure events is almost sure.

By definition, we conclude that g(Xn ) converges to g(X) almost surely.

See also [edit]

  • Slutsky's theorem
  • Portmanteau theorem
  • Pushforward measure

References [edit]

  1. ^ Mann, H. B.; Wald, A. (1943). "On Stochastic Limit and Order Relationships". Annals of Mathematical Statistics. 14 (3): 217–226. doi:10.1214/aoms/1177731415. JSTOR 2235800.
  2. ^ Amemiya, Takeshi (1985). Advanced Econometrics. Cambridge, MA: Harvard University Press. p. 88. ISBN0-674-00560-0.
  3. ^ Sargan, Denis (1988). Lectures on Advanced Econometric Theory. Oxford: Basil Blackwell. pp. 4–8. ISBN0-631-14956-2.
  4. ^ Billingsley, Patrick (1969). Convergence of Probability Measures. John Wiley & Sons. p. 31 (Corollary 1). ISBN0-471-07242-7.
  5. ^ Van der Vaart, A. W. (1998). Asymptotic Statistics. New York: Cambridge University Press. p. 7 (Theorem 2.3). ISBN0-521-49603-9. {{cite book}}: CS1 maint: ref duplicates default (link)

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Source: https://en.wikipedia.org/wiki/Continuous_mapping_theorem

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