\input style \chapnotrue\chapno=3\subchno=3\subsubchno=1 \dfn{ ~$F_n(x)$:} $$ F_n(x)={\hbox{ $X_1$, $X_2$,~\dots, $X_n$, ~$\le x$} \over n}. \eqno(10) $$ .~4 ( , , ~$F_n(x)$). ‌ ~$F(x)$. ~$n$ ~$F_n(x)$ ~$F(x)$. ---ገ (-) , ~$F(x)$ . \emph{ ~$F(x)$ ~$F_n(x)$.} , ~$F(x)$. .~4,~b , ~$X_i$ , . .~4,~c ; , ~$F_n(x)$ ~$F(x)$ ; - , . : $$ \eqalign{ K_n^+&=\sqrt{n}\max_{-\inftyF$, ~$K_n^-$--- ~$F_n30$ , ~$n=\infty$)} } & p=99\% & p=95\% & p=75\% & p=50\% & p=25\% & p=5\% & p=1\%\cr n=1 & 0.01000 & 0.05000 & 0.2500 & 0.5000 & 0.7500 & 0.9500 & 0.9900\cr n=2 & 0.01400 & 0.06749 & 0.2929 & 0.5176 & 0.7071 & 1.0980 & 1.2728\cr n=3 & 0.01699 & 0.07919 & 0.3112 & 0.5147 & 0.7539 & 1.1017 & 1.3589\cr n=4 & 0.01943 & 0.08789 & 0.3202 & 0.5110 & 0.7642 & 1.1304 & 1.3777\cr n=5 & 0.02152 & 0.09471 & 0.3249 & 0.5245 & 0.7674 & 1.1392 & 1.4024\cr n=6 & 0.02336 & 0.1002 & 0.3272 & 0.5319 & 0.7703 & 1.1463 & 1.4144\cr n=7 & 0.02501 & 0.1048 & 0.3280 & 0.5364 & 0.7755 & 1.1537 & 1.4246\cr n=8 & 0.02650 & 0.1086 & 0.3280 & 0.5392 & 0.7797 & 1.1586 & 1.4327\cr n=9 & 0.02786 & 0.1119 & 0.3274 & 0.5411 & 0.7825 & 1.1624 & 1.4388\cr n=10 & 0.02912 & 0.1147 & 0.3297 & 0.5426 & 0.7845 & 1.1658 & 1.4440\cr n=11 & 0.03028 & 0.1172 & 0.3330 & 0.5439 & 0.7863 & 1.1688 & 1.4484\cr n=12 & 0.03137 & 0.1193 & 0.3357 & 0.5453 & 0.7880 & 1.1714 & 1.4521\cr n=15 & 0.03424 & 0.1244 & 0.3412 & 0.5500 & 0.7926 & 1.1773 & 1.4606\cr n=20 & 0.03807 & 0.1298 & 0.3461 & 0.5547 & 0.7975 & 1.1839 & 1.4698\cr n=30 & 0.04354 & 0.1351 & 0.3509 & 0.5605 & 0.8036 & 1.1916 & 1.4801\cr & 0.07089 & 0.1601 & 0.3793 & 0.5887 & 0.8326 & 1.2239 & 1.5174 \cr n>30 & -{0.15\over \sqrt n} & -{0.14\over \sqrt n} & -{0.15\over \sqrt n} & -{0.15\over \sqrt n} & -{0.16\over \sqrt n} & -{0.17\over \sqrt n} & -{0.20\over \sqrt n}\cr } ~$n$, - ~$n$. 䎐~(11) , ! , ~$F(x)$--- , a $F_n(x)$~ , ~$K_n^+$ ~$K_n^-$ : {\sl 考~1.\/} ~$X_1$, $X_2$,~\dots, $X_n$. {\sl 考~2.\/} ~$X_i$ , ~$X_1\le X_2 \le \ldots \le X_n$. (픔 , .~5.) %% 64 {\sl 考~3.\/}~ $$ \eqalign{ K_n^+&=\sqrt{n}\max_{1\le j \le n}\left({j\over n}-F(X_j)\right),\cr K_n^-&=\sqrt{n}\max_{1\le j \le n}\left(F(X_j)-{j-1\over n}\right).\cr } \eqno(13) $$ ᄅ ~$n$ , ~$\chi^2$, . ~$X_j$ ~$F(x)$, - ~$G(x)$, , , ~$G(x)\ne F(x)$; $n$~ , ~$G_n(x)$ ~$F_n(x)$. , ~$n$ .   . \emph{}~$n$. ⎒ $n$~ , , ~$n$. ~$n$ , , $1000$ ~$K_{1000}^+$ : $$ K_{1000}^+(1), \quad K_{1000}^+(2), \quad \ldots, \quad K_{1000}^+(r). \eqno(14) $$ \emph{} \emph{} -. ⅏ $F(x)$--- ~$K_{1000}^+$, ~$F_r(x)$, ~(14). , ~$F(x)$ : ~$n$, $n=1000$, ~$K_n^+$ $$ F_{\infty}(x)=1-e^{-2x^2}, \rem{$x \ge 0$}. \eqno(15) $$ ~$K_n^-$, ~$K_n^+$ ~$K_n^-$ . \emph{  , ~$n$, , - , , .} ( ) . ⅑ " ~5", , ~$1000$ . %% 65 200~ $X_1$, $X_2$,~\dots, $X_{200}$, , ~$F(x)=x^5$ ($0\le x \le 1$). 20~ 10~, ~$K_{10}^+$. 20~~$K_{10}^+$ , .~4; ~$K_{10}^+$. .~4,~a ~$K_{10}^+$, $$ \eqalign{ Y_{n+1}&=(3141592653Y_n+2718281829) \bmod 2^{35},\cr U_n&=Y_n/2^{35}.\cr } $$ 풓 . .~4,~b , 䈁; \emph{} , .~.\ , ~$X_n$ " ~5" ~$F(x)=x^5$. .~4,~c , : $$ Y_{n+1}=((2^{18}+1)Y_n+1)\bmod 2^{35}, \quad U_n=Y_n/2^{35}. $$ - ~(12). .~2 ~$n=20$, , ~$K_{20}^+$ ~$K_{20}^-$ ~(b) ( ~$95\hbox{-}$ ~$12\%\hbox{-}$ ), , . ~$K_{20}^-$ ~(c), , , " ~5" . - , , 10~. 20~ 1000~, ~(b) . - \emph{} , .~4. : $$ \matrix{ & a & b & c \cr K_{200}^+ & 0.477 & 1.537 & 2.819 \cr K_{200}^- & 0.817 & 0.194 & 0.058 \cr } \eqno(16) $$ ⅏ , 䈁. ~(c) ~$n=1\,000\,000$, ~$n=200$ . %% 66 ---ገ , , . $n$~\emph{ } ~$X_1$,~\dots, $X_n$ \emph{} ~$F(x)$. ($F(x)$~ , .~3,~b ~3,~c, .~.\ .~3,~a.) , ~(13), ~$K_n^+$ ~$K_n^-$. 품 .~2. ⅏ ---ገ ~$\chi^2$. , - \emph{ } ~$\chi^2$, , ad hoc, ~$\chi^2$. (풎 , , , , "".) , ~$\chi^2$ , , 10~ , ~$V_1$, $V_2$,~\dots, $V_{10}$. ~$V$--- ( , \emph{} \emph{} , ). 10~ , .~1. ~$K_{10}^+$ ~$K_{10}^-$ ~$\chi^2$. ~10 100~ , ; ~$V$ ~$\chi^2$. , \emph{ 20~ .~4,~c $5\hbox{-}$ ~$95\%\hbox{-}$ ,} \emph{} ; . ---ገ ~$\chi^2$ , , , --- ( $k$~).   , . , ~$\chi^2$ , ~$F(x)$ $k$~ . , , , ~$U_1$, $U_2$,~\dots, $U_n$ %% 67 , .~.\ ~$F(x)=x$ ~$0\le x \le 1$, -. ~$0$ ~$1$ ~$k=100$ , , ~$U$, ~$\chi^2$ 99~ . , . , - , ~$\chi^2$, . , , , ~$(0, 1)$, ~$0$ ~$99$ ~$0$--$49$ ~$50$--$99$, ~$F(x)$, ~$\chi^2$. ~$0$, $2$,~\dots, $98$ ~$1$, $3$,~\dots, $99$, ~$F(x)$. , . $200$~, .~4, ~$\chi^2$ ~$k=10$ ~$V$, ~$9.4$, $17.7$ ~$39.3$; , - (.~(16)). - , ~$\chi^2$ ~$n$. , , .~2, ~$\chi^2$ -. 품 ~$n=200$ " ~$t$" ~$1\le t \le 5$; ~$(0, 1)$ ~$10$~ . ~$K_{200}^+$ ~$K_{200}^-$ , .~2 (, ~$99\%$ ~.~.). 풎 .~5. , ~D ( ), .~5, , \emph{ } ~$\chi^2$ . ~E ( 䈁), , .~5 . 厐 ~A ~B . .~2 ~5 , (a)~ ~$200$ ; (b)~ "", "" " " . ( , %% 68 1948~.\ "" , . ENIAC . , \picture{.~5. - , .~2.} ~$23$ ~$10^8+1$; . ~$23$ , : . , , , , .   ~$23^k$, ~$k$---, !) \section {C. , }. ~$\chi^2$ 1900~. ({\sl Philosophical Magazine,\/} Series~5, {\bf 50}, 157--175). 풀 , , . . %% 68 , ( - 1892~.) , , , ~$10^{29}$. ~$\chi^2$ .~ (W.~G.~Cochran, {\sl Annals of Mathematical Statistics,\/} {\bf 23} (1952), 315--345). , ~$\chi^2$. ⎗ , ~$Y_1=y_1$,~\dots, $Y_k=y_k$ , , $$ {n! \over y_1! \ldots y_k!} p_1^{y_1}\ldots p_k^{y_k}. \eqno(17) $$ , $Y_s$~ $$ {e^{-np_s} (np_s)^{y_s} \over y_s!} $$ ~$Y_s$ , ~$(y_1,~\ldots, y_k)$ $$ \prod_{1\le s \le k} {e^{-np_s} (np_s)^{y_s} \over y_s!}, $$ ~$Y_1+\cdots+Y_k$ ~$n$ $$ \sum_{ \scriptstyle y_1+\cdots+y_k=n \atop \scriptstyle y_1, \ldots, y_k \ge 0 }\prod_{1\le s \le k} {e^{-np_s}(np_s)^{y_s}\over y_s!}={e^{-n}n^n\over n!}. $$ ~$Y_1+\cdots+Y_k=n$, \emph{} , , ~$(Y_1,~\ldots, Y_k)=(y_1,~\ldots, y_k)$, $$ \left(\prod_{1\le s\le k} {e^{np_s}(np_s)^{y_s}\over y_s!}\right) \bigg/ \left({e^{-n}n^n\over n!}\right), $$ ~(17). \emph{, , , ~$Y$ , , .} $$ Z_s={Y_s-np_s \over \sqrt{np_s}},\quad V=Z_1^2+\cdots+Z_k^2. \eqno(18) $$ 㑋~$Y_1+\cdots+Y_k=n$ $$ \sqrt{p_1}Z_1+\cdots+\sqrt{p_k}Z_k=0. \eqno(19) $$ %% 70 $(k-1)\hbox{-}$ ~$S$ ~$(Z_1,~\ldots, Z_k)$, ~(19). ~$n$ ~$Z_s$ (.~.~1.2.10-16), , ~$dZ_2~\ldots{} dZ_k$ ~$S$ \emph{} ~$\exp(-(Z_1^2+\cdots+Z_k^2)/2)$. ( , ~$\chi^2$ , ~$n$.) ⅏ , ~$V\le v$, $$ {\displaystyle\int_{\scriptstyle (Z_1,\ldots, Z_k)\hbox{ } S\atop\scriptstyle\hbox{ } Z_1^2+\cdots+Z_k^2\le v} \exp(-(Z_1^2+\cdots+Z_k^2)/2)\,dz_2\ldots dz_k \over \displaystyle\int_{(Z_1,\ldots, Z_k)\hbox{ } S} \exp(-(Z_1^2+\cdots+Z_k^2)/2) \,dz_2\ldots dz_k }. \eqno(20) $$   ~(19) $k\hbox{-}$ , $(k-1)\hbox{-}$ . ~$\chi$ ~$\omega_1$,~\dots, $\omega_{k-2}$ ~(20) $$ {\displaystyle\int_{\chi^2 \le v} e^{-\chi^2/2}\chi^{k-2} f(\omega_1,~\ldots, \omega_{k-2}) \, d\chi\, d\omega_1\ldots d\omega_{k-2} \over \displaystyle\int e^{-\chi^2/2}\chi^{k-2} f(\omega_1, \ldots, \omega_{k-2})\, d\chi\, d\omega_1\ldots d\omega_{k-2} }; $$ ~$f$ .~15. ~$\omega_1$,~\dots, $\omega_{k-2}$ , . $$ {\displaystyle\int_0^{\sqrt{v}} e^{-\chi^2/2}\chi^{k-2}\, d\chi \over \displaystyle\int_0^\infty e^{-\chi^2/2}\chi^{k-2}\,d\chi }, \eqno(21) $$ , ~$V\le v$. ~$\chi$, ; ~$\chi^2$ . ~$t=\chi^2/2$, -, .~1.2.11.3: $$ \lim_{n\to\infty} P\{V\le v\}=\gamma\left({k-1\over 2}, {v\over2}\right)\bigg/\Gamma\left({k-1\over 2}\right). \eqno(22) $$   ~$\chi^2$ $(k-1)$~ . %% 71 -. 1933~.\ .~.~ , $$ K_n=\sqrt{n}\max_{-\infty=U_0, U_1, U_2, \ldots\,, \eqno(1) $$ . , ~(1). $$ \=Y_0, Y_1, Y_2, \ldots\,, \eqno(2) $$ : $$ Y_n=\floor{d U_n}. \eqno(3) $$ ~$0$ ~$d-1$, ~(1) ~$0$ ~$1$. ~$d$ ; , , , ~$d=64=2^6$; ~$Y_n$ ~$U_n$. 璎 , ~$d$ , , . ~$U_n$, $Y_n$ ~$d$ . ~$d$ . \section{A.~ ( )}. , ~(1), , ~$U_n$ . : (a)~~ ---ገ, ~$F(x)=x$ ~$0\le x \le 1$; (b)~ - ~$d$, ~$100$, , %% 76 , ~$64$ ~$128$ , , ~(2) ~(1). ~$r$, $0\le r < d$, ~$Y_j=r$ ~$0 \le j < n$ ~$\chi^2$ ~$k=d$ ~$p_s=1/d$. . \section{B.~ }. . , ~$(Y_{2j}, Y_{2j+1})=(q, r)$ ~$0 \le j < n$. ~$q$ ~$m$ ~$0$ ~$d$. ~$\chi^2$ ~$k=d^2$ ~$1/d^2$ . ~$d$ , , , ~$\chi^2$ ~$n$, ~$k$ (, ~$n>5d^2$). , , ~.~.\ (.~.~2); ~$d$ , . , , " ~$t$" ( ). , $n$~ $2n$~ ~(2). ~$(Y_0, Y_1)$, $(Y_1, Y_2)$,~\dots, $(Y_{n-1}, Y_n)$; , ? ~$(Y_{2j+1}, Y_{2j+2})$ . , , ({\sl Annals af Mathematical Statistics,\/} {\bf 28}, (1957), 262--264), ~$d$--- , ~$(Y_0, Y_1)$, $(Y_1, Y_2)$,~\dots, $(Y_{n-1}, Y_n)$ $\chi^2\hbox{-}$ ~$V_2$, , \emph{ } ~$V_1$ ~$Y_0$, $Y_1$,~\dots, $Y_{n-1}$ ~$d$, ~$n$ ~$V_2-2V_1$ ~$\chi^2$ $(d-1)^2$~ , ~$V_2$ , \emph{} ~$\chi^2$ $d^2-1$~ . \section{C.~ }. ~$U_j$, . ~$\alpha$ ~$\beta$--- , ~$0\le\alpha<\beta\le 1$, ~$U_j$, $U_{j+1}$,~\dots, $U_{j+r}$, ~$U_{j+r}$ %% 77 ~$\alpha$ ~$\beta$. (  $r+1$~ ~$r$.) \alg G.( .) ዅ ~$0$, $1$,~\dots, $t-1$ ~($\ge t$) ~(1). , $n$~. \picture{ ~ 6. ( ). } \st[ .] 㑒~$j\asg -1$, $s\asg 0$, ~$|COUNT|[r]\asg 0$ ~$0\le r \le t$. \st[$r=0$.] 㑒~$r\asg0$. \st[$\alpha \le U_j < \beta$?] ゅ~$j$ ~$1$. ~$U_j\ge\alpha$ ~$U_j<\beta$, ~\stp{5}. \st[ゅ~$r$.] ゅ~$r$ , ~\stp{3}. \st[ .] ( ~$r$.) ~$r\ge t$ ~$1$ ~$|COUNT|[t]$, ~$1$ ~$|COUNT|[r]$. \st[ $n$~?] ~$1$ ~$s$. ~$s0$, ~\stp{2}. \st[~$f$.] $$ \eqalignno{ f&=C[t]+tC[t-1]+t(t-1)C[t-2]+\cdots+t!C[1]=\cr &=(\ldots((C[1]\times2+C[2])\times3+C[3])\times4+\cdots+C[t-1])\times t+C[t]. \endmark & (7)\cr } $$ \algend , $$ 0\le C[r]X_{j+1}$, $$ \vert 1\, 2\, 9 \vert 8 \vert 5 \vert 3\, 6\, 7 \vert 0\, 4 \vert. \eqno (9) $$ : ~3, ~1, ~3 ~2. .~12 , . %% 82 ( ) \emph{ ~$\chi^2$ ,} \emph{ } . . - ~$\chi^2$ . , , .~12, $$ V={1\over n}\sum_{1\le i, j \le 6} (|COUNT|[i]-nb_i)(|COUNT|[j]-nb_j)a_{ij}, \eqno(10) $$ ~$a_{ij}$ ~$b_i$ : $$ \eqalign{ \pmatrix{ a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16}\cr a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26}\cr a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36}\cr a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46}\cr a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56}\cr a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66}\cr }&= \pmatrix{ 4529.4 & 9044.9 & 13568 & 18091 & 22615 & 27892\cr 9044.9 & 18097 & 27139 & 36187 & 45234 & 55789\cr 13568 & 27139 & 40721 & 54281 & 67852 & 83685\cr 18091 & 36187 & 54281 & 72414 & 90470 & 11580\cr 22615 & 45234 & 67852 & 90470 & 113262 & 139476\cr 27892 & 55789 & 83685 & 111580 & 139476 & 172860\cr },\cr \pmatrix{ b_1 & b_2 & b_3 & b_4 & b_5 & b_6 \cr } &= \pmatrix{ 1\over 6 & 5\over 24 & 11\over 120 & 19 \over 720 & 29 \over 5040 & 1\over 840}.\cr } \eqno(11) $$ ( ; .) \emph{ᒀ~$V$ ~(10) ~$n$ ~$\chi^2$ {\rm ( )} .} ~$n$ , , $4000$ . . .~14, . . $n$~ $Z_{pi}=1$, ~$i$ ~$p$, ~$Z_{pi}=0$ . ~(9) ~$n=10$. $$ Z_{11}=Z_{21}=Z_{31}=Z_{14}=Z_{15}=Z_{16}=Z_{26}=Z_{36}=Z_{19}=Z_{29}=1, $$ ~$Z$ ~$0$. ⎃ $$ R'_p=Z_{p1}+Z_{p2}+\cdots+Z_{pn} \eqno(12 $$ %% 83 ~$p$, $$ R_p=R'_p-R'_{p+1} \eqno(13) $$ , ~$p$. ~$R_p$, \dfn{}% \note{1}{ , .~., 1- (.~.~1.2.10).---{\sl . .\/}} $$ \covar (R_p, R_q)=\mean ((R_p-\mean (R_p)) (R_q-\mean (R_q))), $$ ~$R_p$ ~$R_q$. 㑐 $n!$~. 䎐~(12) ~(13) , ~$Z_{pi}$ ~$Z_{pi}Z_{qj}$, ( , ~$i1$. , ~$Z_{pi}Z_{qj}$ ~$0$, ~$1$, , ~$U_1$, $U_2$,~\dots, $U_n$, ~$Z_{pi}=Z_{qj}=0$, $$ U_{i-1}>U_i<\ldotsU_{i+p}<\ldotsn$;\cr } & (18) \cr } $$ $t=\max(p,q)$, $s=p+q$ $$ \eqalignno{ f(p, q, n) &= (n+1)\left({s(1-pq)+pq\over (p+1)!(q+1)!}-{2s\over (s+1)!}\right)+\cr &+2\left({s-1\over s!}\right)+{(s^2-s-2)pq-s^2-p^2q^2+1\over (p+1)!(q+1)!}. &(19)\cr } $$ , , . $$ \eqalign{ \mean(R_p)&=\mean(R'_p)-\mean(R'_{p+1}),\cr \covar (R_p, R'_q)&=\covar(R'_p, R'_q)-\covar(R'_{p+1}, R'_q),\cr \covar(R_p, R_q)&=\covar(R_p, R'_q)-\covar(R_p, R'_{q+1}).\cr } \eqno(20) $$ .~ ({\sl Annals of Mathematical Statistics,\/} {\bf 15} (1944), 163--165) , ~$n\to\infty$ ~$R_1$, $R_2$,~\dots, $R_{t-1}$, $R'_t$ , . , . $n$~ ~$R_p$--- ~$p$, ~$1\le p < t$, ~$R'_t$--- ~$\ge t$. $$ \eqalign{ Q_1&=R_1-\mean(R_1), \ldots, Q_{t-1}=R_{t-1}-\mean(R_{t-1}),\cr Q_t&=R'_t-\mean(R'_t).\cr } \eqno(21) $$ %% 85 ~$C$, , , $C_{13}=\covar(R_1, R_3)$, ~$C_{1t}=\covar(R_1, R'_t)$. ~$t=6$ $$ \eqalign{ C&= nC_1+C_2=\cr &= n\pmatrix{ 23 \over 180 & -7 \over 360 & -5 \over 336 & -433 \over 60480 & -13 \over 5670 & -121 \over 181440 \cr -7 \over 360 & 2843 \over 20160 & -989 \over 20160 & -7159 \over 362880 & -10019 \over 1814400 & -1303 \over 907200 \cr -5 \over 336 & -989 \over 20160 & 54563 \over 907200 & -21311 \over 1814400 & -62369 \over 19958400 & -7783 \over 9979200 \cr -433 \over 60480 & -7159 \over 362880 & -21311 \over 1814400 & 886657 \over 39916800 & -257699 \over 239500800 & -62611 \over 239500800 \cr -13 \over 5670 & -10019 \over 1814400 & -62369 \over 19958400 & -257699 \over 239500800 & 29874811 \over 5448643200 & -1407179 \over 21794572800 \cr -121 \over 181440 & -1303 \over 907200 & -7783 \over 9979200 & -62611 \over 239500800 & -1407179 \over 21794572800 & 2134697 \over 1816214400 \cr }+\cr &+\pmatrix{ 83 \over 180 & -29 \over 180 & -11 \over 210 & -41 \over 12096 & 91 \over 25920 & 41 \over 18144 \cr -29 \over 180 & -305 \over 4032 & 319 \over 20160 & 2557 \over 72576 & 10177 \over 604800 & 413 \over 64800 \cr -11 \over 210 & 319 \over 20160 & -58747 \over 907200 & 19703 \over 604800 & 239471 \over 19958400 & 39517 \over 9979200 \cr -41 \over 12096 & 2557 \over 72576 & 19703 \over 604800 & -220837 \over 4435200 & 1196401 \over 239599800 & 360989 \over 239500800 \cr 91 \over 25920 & 10177 \over 604800 & 239471 \over 19958400 & 1196401 \over 239500800 & -139126639 \over 7264857600 & 4577641 \over 10897286400 \cr 41 \over 18144 & 413 \over 64800 & 39517 \over 9979200 & 360989 \over 239500800 & 4577641 \over 10897286400 & -122953057 \over 21794572800 \cr }.\cr } \eqno(22) $$ ~$n\ge 14$. ~$A=(a_{ij})$, ~$C$, ~$\sum_{1\le i,j \le t} Q_i Q_j a_{ij}$. ~$n$ ~$\chi^2$ ~$t$~ . ~(11)--- ~$C_1$, . ~$n$ ~$A$ ~$(1/n)C_1^{-1}$. , ~$C_1$ , ~$t=4$. ~$n=1000$ ~(11) ~1\% , ~(22). ᒀ .~2.2.6, .~18. %%86 \section{H.~⅑ " ~$t$"}. ~$V_j=\max(U_{tj}, U_{tj+1},~\ldots, U_{tj+t-1})$ ~$0\le j < n$. -ገ ~$V_0$, $V_1$,~\dots, $V_{n-1}$, ~$F(x)=x^t$, ($0\le x \le 1$). ~$V_0^t$, $V_1^t$,~\dots, $V_{n-1}^t$. , $V_j$~ ~$F(x)=x^t$. , ~$\max(U_1, U_2,~\ldots, U_t)\le x$, , ~$U_1\le x$ \emph{}~$U_2\le x$ \emph{}~\dots{} \emph{}~$U_t\le x$; , ~$x\cdot x \cdot \ldots \cdot x = x^t$. \section{I.~ }. $$ C={n(U_0U_1+\cdots+U_{n-2}U_{n-1}+U_{n-1}U_0)-(U_0+\cdots+U_{n-1})^2 \over n(U_0^2+\cdots+U_{n-1}^2)-(U_0+\cdots+U_{n-1})^2}. \eqno(23) $$ 풎 " ", ~$U_{j+1}$ ~$U_j$. ~$n$ , , ~$C$ (L.~P.~Schmid, {\sl CACM,\/} {\bf 8} (1965), 115). . ~$U_0$, $U_1$,~\dots, $U_{n-1}$ ~$V_0$, $V_1$,~\dots, $V_{n-1}$, : $$ C={n\sum(U_j V_j)-(\sum U_j)(\sum V_j) \over \sqrt{(n\sum U_j^2 - (\sum U_j)^2) (n\sum V_j^2 - (\sum V_j)^2)}}. \eqno(24) $$ ᓌ ~$j$ ~$0\le j < n$. 䎐~(23) ~$V_j=U_{(j+1)\bmod n}$. [\emph{.} ~(24) ~$U_0=U_1=\cdots=U_{n-1}$ ~$V_0=V_1=\cdots=V_{n-1}$; .) ~$-1$ ~$+1$. , ~$U_j$ ~$V_j$, ~$\pm 1$, , .~.\ ~$j$ ~$V_j=m \pm aU_j$ ~$a$ ~$m$ (.~.~17).   , , ~$C$, ~(23), . , - , ~$U_0U_1$ ~$U_1U_2$, , , \emph{} (.~.~18). "厐" ~$C$, ~$\mu_n-2\sigma_n$ ~$\mu_n+2\sigma_n$, $$ \mu_n={-1\over (n-1)}, \quad \sigma_n={1\over n-1}\sqrt{n(n-3)\over n+1}, \rem{$n>2$.} \eqno(25) $$ %% 87 ~$C$ 95\% . 䎐~(25) , ~$C$ , $U$~ , . ዓ ~$U$ .~ ({\sl Annals of Mathematical Statistics,\/} {\bf 15} (1944), 119--144). , , , .~.~~(25), . , ~$\lim_{n\to\infty}\sqrt{n}\sigma_n=1$; .~ ㎊ ({\sl Annals of Mathematical Statistics,\/} {\bf 35} (1964), 1296--1303), \emph{} . \section{J.~ }. , . , ~$X$, $Y$ ~$Z$, . , , \emph{} . $q$~, , , ~$U_0$, $U_1$, $U_2$,~\dots, $$ U_0, U_q, U_{2q}, \ldots; \quad U_1, U_{q+1}, U_{2q+1}, \ldots; \quad \ldots; \quad U_{q-1}, U_{2q-1},\ldots\,. \eqno(26) $$ , , , , ~$q$ , , .  , ~$m$, , ~$q<16$ ~$q=8$; ~$q=10$. (풎 , , ~$q$ , , .) \section{K.~ }. ᒀ %% 88 , "" "" - , . , , .~ ~.~-ገ [{\sl Journal of the Royal Statistical Society,\/} {\bf 101} (1938), 147--166, , {\bf 6} (1939), 51--61]. ~$0$ ~$9$, ; , , , - ( ). -ገ , , , 1955~. ⅑ . , ( ). ዅ , , , , ~$U$ ; , , ~$U_i=U_j$, ~$i\ne j$, . [J.~Bienaynie, {\sl Comptes Rendus,\/} {\bf 81} (Paris: Acad.\ Sciences, 1875), 417--423]. 60~ - , [{\sl Proc.\ Royal Society Edinburgh,\/} {\bf 57} (1937), 228--240, 332--376]. , 턈 ~1785~.\ ~1930~.\ ( ). , ~1944~.\ , ~$\chi^2$ . [{\sl Annals of Mathematical Statistics,\/} {\bf 15} (1944), 58--69] ( ) . , , , , , (.~ Barton~D.~E., Mallows~.~L., {\sl Annals of Math.\ Statistics,\/} {\bf 36} (1965), 236--260]. --- , , %% 89 . ~\S~3.5 (.~.~3.5-26). : .~3.3.3-23, 24 , ~$m$ , , , . , : \emph{" ?".} , ", ! 풎 , . . , , , , , , , . , , , ~$m$ , , . ⅑ " ~$t$" , . , , , . - , , . , \emph{,} . , , , ; , - , , . \excercises \ex[10] (.~.~) ~$(Y_0, Y_1)$ $(Y_2, Y_3)$,~\dots, $(Y_{2n-2}, Y_{2n-1})$, ~$(Y_0, Y_1)$, $(Y_1, Y_2)$,~\dots, $(Y_{n-1}, Y_n)$? \ex[10] , , ~.~. \rex[20] ኎ ~$U$ (~G), $n$~, %% 90 , ? ? \ex[12] , ~(4). \ex[M23] "" , -ገ, $N$~~$U$, , , $U_{N+j}$~ ~$U_j$. $n$~ ~$U_0$,~\dots, $U_{N-1}$ ~$\alpha\le U_j < \beta$, $n$~. ~$Z_r$--- ~$r$, ~$0\le rX_{j+1}$ ~$X_{j+2}$, ~$\chi^2$ ( , ). ? \ex[M20] ~$V_0^t$, $V_1^t$,~\dots, $V_{n-1}^t$ " ~$t$" ? \rex[15] (a)~ " ~$t$" ~$t$. ~$Z_{jt}=\max (U_j, U_{j+1},~\ldots, U_{j+t-1})$. ᒓ ጛ ~$Z_{0(t-1)}$, %% 91 $Z_{1(t-1)}$,~\dots{} ~$Z_{0t}$, $Z_{1t}$,~\dots, . . (b)~ " ~$t$" , ~$V_j=\max(U_j,~\ldots, U_{j+t-1})$; , $V_j=Z_{jt}$, %% ?? Z_{(t_j)t} ~$V_j=Z_{(tj)t}$, . : \emph{} , , ~$Z_{jt}$, $0\le j h$, ~$c$ ~$c\bmod h$ ~(30) ~C. {\sl 考~5.\/}~⅏ ~B, $$ \sigma(h, k, c)=-3+{h\over k}+{k\over h}+{1+6c^2\over hk}-\sigma(k, h, c). \eqno (35) $$ ~$\sigma(k, h, c)$ ~1. 爑 ; ~$h$ ~$k$ , , (.~.~4.5.2) ~$h$ ~$k$. . \proclaim ~1. ~$m=2^{35}$, $a=2^{34}+1$, $c=1$. \solution ᎃ~(17), $$ C=(2^{35}\sigma(2^{34}+1, 2^{35}, 1)-3+6(2^{35}-(2^{34}-1)-1))/(2^{70}-1). \eqno (36) $$ ~1 ~2, $$ (\sigma(2^{34}+1, 2^{35}, 1)=-\sigma(2^{34}-1, 2^{35}, 1). $$ ᎃ ~5: $$ \sigma(2^{34}-1, 2^{35}, 1)=-3+(2^{34}-1)/2^{35}+2^{35}/(2^{34}-1) +7/2^{35}(2^{34}-1)-\sigma(2^{35}, 2^{34}-1, 1). $$ ᎃ ~1: $$ \sigma(2^{35}, 2^{34}-1, 1)=\sigma(2, 2^{34}-1, 1). $$ ⅏ ~5 $$ \sigma(2, 2^{34}-1, 1)=-3+2/(2^{34}-1)+(2^{34}-1)/2 +7/2(2^{34}-1)-\sigma(2^{34}-1, 2, 1) $$ $$ \sigma(2^{34}-1, 2, 1)=0. $$ , $$ C={1\over 4}+\varepsilon, \rem{$\abs{\varepsilon}<2^{-67}$.} \eqno(37) $$   , , . , ; . \proclaim ~2. ~$m=10^{10}$, $a=10001$, $b=2113248658$. %% 100 \solution   ~$C\approx \sigma(a, m,c)/m$, : \EQ[38]{ \eqalign{ \sigma(10001, 10^{10}, 2113248653) &= \sigma(10001, 10^{10}, 7350)-6(211303)(7886743997)/10^{10};\cr \sigma(10001, 10^{10}, 7350)&\approx -3+10^{10}/10001-\sigma(10^{10}, 10001, 7350);\cr \sigma(10^{10}, 10001, 7350)&=\sigma(100, 10001, 7350)=\cr &=\sigma(100, 10001, 50)-6(73)(2601)/10001;\cr \sigma(100, 10001, 50)&\approx -3+10001/100+100/10001-\sigma(10001, 100, 50);\cr \sigma(10001, 100, 50)&=\sigma(1, 100, 50)=-50,02.\cr C&\approx(-3+999900,01-97,02-50,02+113,91-99895,60)/10^{10}=\cr &=-0,000000003172.\cr } }   ~$C$, , . ~$3$, \emph{ , , .} --- , ! \proclaim ~3. ~$a$, $m$, $c$. . ~$c_0=c\bmod a$. $$ \eqalignno{ \sigma(a, m, c)&=\sigma(a, m, c_0)+{6(c-c_0)\over am}(c+c_0-m)=\cr &=-3+{a\over m}+{m\over a}+{1\over am}+{6c^2\over am}-{6(c-c_0)\over a}-\sigma(m, a, c_0).&(39)\cr } $$ ᎃ .~12, $\abs{\sigma(m, a, c_0)}