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My question is "how weak convergences of measurable functions is defined?" There seems to be two different definitions which are both based on weak convergence of measures generated by the measurable functions, but differ in how the measurable functions generate their new measures.

  1. In terms of the new measure $\nu(A):=\int_A f \,d\mu$ generated from a measurable function $f$ wrt a measure $\mu$ on its domain:

    From "Weak Compactness" in Section 19 "The $L^p$ spaces" of "Probability and Measure" by Billingsley:

    Suppose that $f$ and $f_n$ are elements of $L^p(\Omega, \mathcal{F},\mu)$. If $\int f \times g \, d\mu= \lim_{n\rightarrow \infty} \int f \times g \, d\mu$ for each $g$ in $L^q(\Omega, \mathcal{F},\mu)$ with $1/p + 1/q = 1$, then $f_n$ converges weakly to $f$.

    At the end of Section 4.1 of "A Course in Probability Theory" by Kai Lai Chung, weak convergence of random variables in $L^1$ space is also defined similarly to Billingsley's.

  2. In terms of the pushforward measure of a measurable function:

    In Wikipedia,

    In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements $\{X_n\}$ converges weakly to $X$ if $$ \operatorname{E}^*h(X_n) \to \operatorname{E}\,h(X) $$ for all continuous bounded functions $h(·)$. Here $E^*$ denotes the outer expectation, that is the expectation of a “smallest measurable function g that dominates $h(X_n)$”.

    Also from Wikipedia:

    Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\textbf{X}$ be a metric space. If $X_n, X: Ω → \textbf{X}$ is a sequence of random variables then $X_n$ is said to converge weakly (or in distribution or in law) to $X$ as $n → ∞$ if the sequence of pushforward measures $(X_n)_∗(P)$ converges weakly to $X_∗(P)$ in the sense of weak convergence of measures on $\textbf{X}$.

I wonder if the two definitions of weak convergence of measurable functions are equivalent? Why are there two different definitions for the same concept?

Thanks and regards!

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1 Answer 1

up vote 3 down vote accepted

In the chapter you quote, Billingsley defines weak convergence of functions in $L^p$., which is by definition the convergence against every function in $L^q$, the dual of $L^p$. That is, $(f_n)$ in $L^p$ converges to $f$ in $L^p$ if and only if $\int f_ng\to\int fg$ for every $g$ in $L^q$ and this is a weak convergence because $L^q=(L^p)^*$.

In the chapter you quote, Chung defines weak convergence of functions in $L^1$, which is by definition the convergence against every function in $L^\infty$, the dual of $L^1$. That is, $(f_n)$ in $L^1$ converges to $f$ in $L^1$ if and only if $\int f_ng\to\int fg$ for every $g$ in $L^\infty$ and this is a weak convergence because $L^\infty=(L^1)^*$.

In the page you quote, Wikipedia defines weak convergence of (probability) measures. This mode of convergence should be called weak* rather than weak because it refers to the convergence against any bounded continuous functions, and the space $M_1$ of probability measures is included in the dual of the space $C_b$ of bounded continuous functions. That is, $(\mu_n)$ in $M_1$ converges to $\mu$ in $M_1$ if and only if $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ for every $f$ in $C_b$ and this is a weak* convergence because $M_1\subset(C_b)^*$.

To mean that the distributions of some random variables converge weakly in the sense above, one usually says that the random variables converge in distribution.

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(2) If I am right, "the random variables converge in distribution" is defined as their cumulative distribution functions converges where they are continuous. Does "to mean that the distributions of some random variables converge weakly in the sense above, one usually says that the random variables converge in distribution", mean that the weak convergence defined by Billingsley and by Chung, and the weak* convergence defined by Wikipedia, are equivalent to each other, and also to convergence of cdfs at their continuity? –  Tim Oct 19 '11 at 12:50
    
(3) I wonder if there is some mistake in my understanding that: the difference between the convergence of Billingsley and Chung and Wikipedia can be seen as the differences in the way the new measures generated by the measurable functions and the original measure; what is common between them is that they are both based on the weak* convergence of the new measures. –  Tim Oct 19 '11 at 14:03
    
Thanks! I am slow to understand your reply. Would you mind explaining what is the continuous dual of $C_b$? I opened a new post here math.stackexchange.com/questions/309783/…. Thanks! –  Tim Feb 22 '13 at 14:58

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