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Next: $B$3$NJ8=q$K$D$$$F(B... (PDF ե: central.pdf)

$BJ?@.(B 13 $BG/(B 6 $B7n(B 22 $BF|(B
$BCf?46K8BDjM}$N>ZL@(B
$B?73c9)2JBg3X(B $B>pJsEE;R9)3X2J(B $BC]LnLP<#(B

$B0lHL$NO"B3J,I[$KBP$9$kCf?46K8BDjM}(B:

$x_1,x_2,\ldots,x_n,\ldots$ $B$,FHN)F1J,I[$G!"$=$NJ?6Q$r(B $\mu$, $BJ,;6$r(B $\sigma^2$ $B$H$9$k$H$-!"(B

\begin{displaymath}
\bar{x}_n=\frac{x_1+x_2+\cdots x_n}{n},\hspace{0.5zw}
y_n=\sqrt{n}\frac{\bar{x}_n-\mu}{\sigma}
\end{displaymath}

$B$H$9$k$H(B $y_n$ $B$NJ,I[$O(B $n\rightarrow\infty$ $B$N$H$-(B $BI8=`@55,J,I[(B $N(0,1)$ $B$K<}B+$9$k!#$9$J$o$A!"(B$n$ $B$,Bg$-$$$H$-!"(B $\bar{x}_n$ $B$NJ,I[$O(B $N(\mu,\sigma^2/n)$ $B$G6a;w$G$-$k!#(B
$B$N>ZL@$O$+$J$jLLE]$G$"$k$,(B ($B>v$_9~$_@QJ,!"$=$N%U!<%j%(JQ49!"(B $n$ $B=E@QJ,$NN`;w@QJ,JQ49$J$I$N=`Hw$,I,MW(B)$B!"(B $B$B%b%"%V%k(B$=$$B%i%W%i%9$NDjM}$J$i$P(B $B%9%?!<%j%s%0$N8x<0$r;H$C$FF3$-=P$9$3$H$,$G$-$k!#(B


$BDjM}(B 1

$0<p<1$, $u$ $B$KBP$7$F(B $\mu=np$, $\sigma=\sqrt{npq}$ ($q=1-p$) $B$H$7!"(B $x=\sigma u+\mu$ ( $u=(x-\mu)/\sigma$) $B$H$9$k$H$-!"(B $p$, $u$ $B$r8GDj$7$?$^$^(B $n\rightarrow\infty$ $B$H$9$k$H(B

\begin{displaymath}
\sigma\left(\begin{array}{c} n \\ x \end{array}\right)p^xq^{n-x}\rightarrow\frac{1}{\sqrt{2\pi}}e^{-u^2/2}
\end{displaymath}

$B$H$J$k!#$h$C$F!"(B$n$ $B$,Bg$-$$$H$-!"Fs9`J,I[(B $B(n,p)$ $B$O(B $B@55,J,I[(B $N(\mu,\sigma^2)$ ($\mu=np$, $\sigma=\sqrt{npq}$) $B$G(B $B6a;w$G$-$k$3$H$K$J$k!#(B


$B$3$N>ZL@$K$O!"%9%?!<%j%s%0$N8x<0(B:

\begin{displaymath}
n!\sim n^n\sqrt{2\pi n}e^{-n}\hspace{0.5zw}(n\rightarrow\infty)\end{displaymath} (1)

$B$rMQ$$$k!#$J$*!"$3$N(B $\sim$ $B$N0UL#$ON>JU$NHf$,(B 1 $B$K<}B+$9$k$3$H$r(B $B0UL#$9$k!#$^$?!"0J2<$G$O%i%s%@%&$N5-9f$H8F$P$l$k(B $O$ ($B%i!<%8%*!<(B), $o$ ($B%9%b!<%k%*!<(B) $B$b;HMQ$9$k$N$G!"$^$:!"$=$N@bL@$r$7$F$*$/!#(B


$BDj5A(B 2

$n\rightarrow\infty$ $B$N$H$-!"(B$a_n=o(b_n)$ ($B%9%b!<%k%*!<(B) $B$O!"(B

\begin{displaymath}
\frac{a_n}{b_n}\rightarrow 0
\end{displaymath}

$B$H$J$k$3$H$r0UL#$7!"(B$a_n=O(b_n)$ ($B%i!<%8%*!<(B) $B$O!"(B$a_n/b_n$ $B$,(B $BM-3&$G$"$k$3$H$r0UL#$9$k!#(B


$a_n=o(b_n)$ $B$N$H$-$O!"(B$a_n$ $B$O(B $b_n$ $B$h$j>.$5$$$b$N!"(B $a_n=O(b_n)$ $B$N$H$-$O!"(B$a_n$ $B$O(B $b_n$ $B$h$j$[$\F1Ey!"$H$$$&$3$H$K$J$k!#(B $BNc$($P(B $a_n\rightarrow 0$ $B$O$3$N5-9f$r;H$($P(B $a_n=o(1)$ $B$H=q$1!"(B $B%9%?!<%j%s%0$N8x<0$O!"$3$N5-9f$r;H$($P(B

\begin{displaymath}
n!= O(n^n\sqrt{2\pi n}e^{-n}),\hspace{1zw}
n!= n^n\sqrt{2\pi n}e^{-n}(1+o(1))
\end{displaymath}

$B$N$h$&$K=q$1$k$3$H$K$J$k!#(B

$BDjM}(B 1 $B$N>ZL@(B

$x=u\sigma+\mu=u\sqrt{npq}+np$ $B$K$h$j(B $x$ $B$b(B $n$ $B$K$h$C$FJQ$o$k$3$H$K(B $BCm0U$9$k!#$^$?!"(B$0<p<1$ $B$H$9$k!#(B

\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri...
... npq + \log n! - \log x! - \log (n-x)!
+ x\log p + (n-x)\log q
\end{eqnarray*}



$B$H$J$k!#(B $B:#!"(B$p>0$, $q>0$ $B$J$N$G!"(B$x$, $n-x$ $B$O(B $n\rightarrow\infty$ $B$N$H$-(B

\begin{eqnarray*}x & = & np+u\sqrt{npq} = n\left(p+u\sqrt{\frac{pq}{n}}\right)
...
...pq}
= n\left(q-u\sqrt{\frac{pq}{n}}\right)
\rightarrow\infty
\end{eqnarray*}



$B$H$J$k!#$^$?!"%9%?!<%j%s%0$N8x<0$h$j(B $m\rightarrow\infty$ $B$N$H$-$O(B

\begin{eqnarray*}\log m! & = & \log m^m\sqrt{2\pi m}e^{-m}(1+o(1)) \\
& = & m\...
...frac{1}{2}\log 2\pi + \left(m+\frac{1}{2}\right)\log m -m +o(1)
\end{eqnarray*}



$B$G$"$k$N$G!"(B

\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri...
...frac{n}{n-x}
+x\log p\frac{n}{x}+(n-x)\log q\frac{n}{n-x}+o(1)
\end{eqnarray*}



$B$H$J$k!#$3$3$G!">e$G8+$?$h$&$K(B

\begin{displaymath}
\frac{x}{n} = p + u\sqrt{\frac{pq}{n}},\hspace{1zw}
\frac{n-x}{n} = q - u\sqrt{\frac{pq}{n}}
\end{displaymath}

$B$G$"$k$N$G!"(B

\begin{displaymath}
\log p\frac{n}{x} \rightarrow \log 1 = 0,\hspace{1zw}
\log q\frac{n}{n-x} \rightarrow \log 1 = 0
\end{displaymath}

$B$H$J$k$N$G(B

\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri...
...p}}\right)
-(n-x)\log \left(1-u\sqrt{\frac{p}{nq}}\right)+o(1)
\end{eqnarray*}



$B$H$J$k!#%F%$%i!

\begin{displaymath}
\log(1+x)=x-\frac{x^2}{2}+O(x^3)\hspace{1zw}(x\rightarrow 0)
\end{displaymath}

$B$rMQ$$$k$H!"(B

\begin{eqnarray*}\lefteqn{x\log\left(1+u\sqrt{\frac{q}{np}}\right)
= (np+u\sqrt...
...-u\sqrt{npq}+\frac{u^2}{2}p+O\left(\frac{1}{\sqrt{n}}\right)\\
\end{eqnarray*}



$B$f$($K(B

\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri...
...\log 2\pi -\frac{u^2}{2}
= \log\frac{1}{\sqrt{2\pi}}e^{-u^2/2}
\end{eqnarray*}







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Next: $B$3$NJ8=q$K$D$$$F(B...
Shigeharu TAKENO
2001$BG/(B 8$B7n(B 9$BF|(B