{"id":1518,"date":"2020-06-29T11:00:00","date_gmt":"2020-06-29T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1518"},"modified":"2020-06-27T20:00:57","modified_gmt":"2020-06-27T11:00:57","slug":"lyapunov-inequality","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/lyapunov-inequality","title":{"rendered":"\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f\u3010\u5f0f\u3068\u8a3c\u660e\u3092\u7d39\u4ecb\uff01\u3011"},"content":{"rendered":"\n

\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2605\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n

\u3000\u30ea\u30e4\u30d7\u30ce\u30d5\uff08\u30ea\u30a2\u30d7\u30ce\u30d5\uff09\u306e\u4e0d\u7b49\u5f0f\u306f\u3001\u3042\u307e\u308a\u6709\u540d\u306a\u5f0f\u3067\u306f\u306a\u304f\u30de\u30cb\u30a2\u30c3\u30af\u306a\u90e8\u985e\u306e\u4e0d\u7b49\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002\u7d71\u8a08\u5b66\u3092\u7814\u7a76\u3057\u3066\u3044\u304f\u4e0a\u3067\u3001\u8a08\u7b97\u65b9\u6cd5\u306e\u624b\u6bb5\u306e\u3072\u3068\u3064\u3068\u3057\u3066\u3001\u77e5\u3063\u3066\u304a\u304f\u3068\u4fbf\u5229\u306b\u611f\u3058\u308b\u3053\u3068\u304c\u3042\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n\n\n\n

\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f<\/h3>\n\n\n\n
\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f(Lyapunov inequality)<\/strong><\/div>\n
\u78ba\u7387\u5909\u6570\\(X\\)\u306b\u95a2\u3057\u3066\u3001\\(\\mathrm{E}[|X|^{p}]<\\infty\\)\u3092\u6e80\u305f\u3059\u3082\u306e\u3068\u3057\u307e\u3059\uff08\u3064\u307e\u308a\u6709\u9650\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\uff09\u3002\u3053\u306e\u3068\u304d\\( 0 < q < p \\)\u306b\u5bfe\u3057\u3066$$\\mathrm{E}[|X|^{q}]^{\\frac{1}{q}}\\leq \\mathrm{E}[|X|^{p}]^{\\frac{1}{p}}$$\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u306e\u5f0f\u3092\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f\u3068\u8a00\u3044\u307e\u3059\u3002\u3053\u306e\u5f0f\u3092\u7528\u3044\u308b\u3068
\\begin{align}\n\\mathrm{E}[|X|] \\leq \\mathrm{E}[|X|^{2}]^{\\frac{1}{2}}\\leq \\mathrm{E}[|X|^{3}]^{\\frac{1}{3}}\\leq \\cdots \\leq\\mathrm{E}[|X|^{n}]^{\\frac{1}{n}}\n\\end{align}<\/div>\u3068\u3044\u3046\u5f0f\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/div>\n\n\n\n

\u3000<\/p>\n\n\n\n

<\/div>

\u3000\u4e0d\u7b49\u5f0f\u306e\u8a55\u4fa1\u306f\u4ee5\u4e0b\u306b\u305f\u304f\u3055\u3093\u306e\u77e5\u8b58\u3092\u6301\u3063\u3066\u3044\u308b\u304b\u3067\u3001\u5f97\u610f\u304b\u4e0d\u5f97\u610f\u304b\u5de6\u53f3\u3055\u308c\u307e\u3059\u3002
\u3000\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f\u306f\u30de\u30cb\u30a2\u30c3\u30af\u306a\u5f0f\u3067\u3059\u304c\u3001\u3072\u3068\u3064\u306e\u6b66\u5668\u3068\u3057\u3066\u662f\u975e\u899a\u3048\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>

<\/div><\/div><\/div>\n\n\n\n

\u8a3c\u660e<\/h4>\n\n\n\n

\u3000\u30d8\u30eb\u30c0\u30fc\u306e\u4e0d\u7b49\u5f0f<\/a>\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u30d8\u30eb\u30c0\u30fc\u306e\u4e0d\u7b49\u5f0f\u306f\u78ba\u7387\u5909\u6570\\(X,\\ Y\\)\u306b\u3064\u3044\u3066$$\\mathrm{E}[|X|^{t}]<\\infty,\\ \\ \\ \\mathrm{E}[|X|^{q}]<\\infty$$\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3082\u306e\u3068\u3057\u307e\u3059\u3002\u3053\u3053\u3067\\(q,\\ t\\)\u306f\\(1\\)\u3088\u308a\u5927\u304d\u3044\u6b63\u306e\u5b9f\u6570\u3067$$\\frac{1}{q}+\\frac{1}{t}=1$$\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001

\\begin{align}
\\mathrm{E}[|XY|] &= \\mathrm{E}[|X|^{q}]^{\\frac{1}{q}}\\mathrm{E}[|Y|^{t}]^{\\frac{1}{t}}
\\end{align}<\/div>
\u304c\u6210\u7acb\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u3053\u3067\u78ba\u7387\u5909\u6570\\(Y\\)\u3092\\(Y=1\\)\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u3092\\(X=X^{q}\\)\u3068\u66f8\u304d\u63db\u3048\u308b\u3068
\\begin{align}
\\mathrm{E}[|X|^{q}] &= \\mathrm{E}[|X|^{tq}]^{\\frac{1}{t}} \\\\
\\end{align}<\/div>
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\\(p=tq\\)\u3067\u7f6e\u304d\u63db\u3048\u308b\u3068\\(q< p\\)\u3068\u306a\u308a\u307e\u3059\u3002\\(\\displaystyle\\frac{1}{t}=\\frac{q}{p}\\)\u3068\u306a\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\u3001
\\begin{align}
\\mathrm{E}[|X|^{q}] &= \\mathrm{E}[|X|^{p}]^{\\frac{q}{p}} \\\\
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u4e21\u8fba\u3092\\(\\displaystyle\\frac{1}{q}\\)\u4e57\u3059\u308b\u3068\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n\n\n\n

\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f\u3068\u7d76\u5bfe\u30e2\u30fc\u30e1\u30f3\u30c8<\/h3>\n\n\n\n

\u30ea\u30e4\u30d7\u30ce\u30d5\u306e\u4e0d\u7b49\u5f0f\u304b\u3089\u7d76\u5bfe\u30e2\u30fc\u30e1\u30f3\u30c8<\/a>\u3067\u306f\u6b21\u306e\u3088\u3046\u306a\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002

\\begin{align}
\\upsilon_{1}^{\\prime}(a) \\leq \\upsilon_{2}^{\\prime}(a)^{\\frac{1}{2}} \\leq \\upsilon_{3}^{\\prime}(a)^{\\frac{1}{3}} \\leq \\cdots \\leq \\upsilon_{n}^{\\prime}(a)^{\\frac{1}{n}}
\\end{align}<\/div>
\u3053\u3053\u3067\u3001\\(\\upsilon_{k}^{\\prime}(a)\\)\u306f\\(a\\)\u306e\u307e\u308f\u308a\u306e\\(k\\)\u6b21\u306e\u7d76\u5bfe\u30e2\u30fc\u30e1\u30f3\u30c8\u3092\u8868\u3057\u307e\u3059\u3002
<\/p>\n","protected":false},"excerpt":{"rendered":"

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