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Revision difference : convergence in distribution |
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| A sequence of distribution functions $F_1,F_2,\dots$ converges \emph{weakly} to a distribution function $F$ if $F_n(t)\rightarrow F(t)$ for each point $t$ at which $F$ is continuous. |
A sequence of distribution functions $F_1,F_2,\dots$ converges \emph{weakly} to a distribution function $F$ if $F_n(t)\rightarrow F(t)$ for each point $t$ at which $F$ is continuous. |
| If the random variables $X,X_1,X_2,\dots$ have associated distribution functions $F,F_1,F_2,\dots$, respectively, then we say that $X_n$ converges \emph{in distribution} to $X$, and denote this by $X_n\xrightarrow[]{D} X$. |
If the random variables $X,X_1,X_2,\dots$ have associated distribution functions $F,F_1,F_2,\dots$, respectively, then we say that $X_n$ converges \emph{in distribution} to $X$, and denote this by $X_n\xrightarrow[]{D} X$. |
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This definition holds for joint distribution functions and random vectors as well.
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This is probably the weakest type of convergence of random variables, but it has showed to be very useful. Some results involving this type of convergence are the central limit theorems, Helly-Bray theorem, Paul-L\'evy continuity theorem, Cram\'er-Wold theorem and Scheff\'e's theorem.
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| This is probably the weakest \PMlinkescapetext{type} of convergence of random variables, but it has showed to be very useful. Some results involving this \PMlinkescapetext{type} of convergence are the central limit theorems, Helly-Bray theorem, Paul-L\'evy continuity theorem, Cram\'er-Wold theorem and Scheff\'e's theorem. |
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