On the law of large numbers for discrete Fourier transform

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• 作者：Na Zhang ^{zhangn4@mail.uc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 60F15 ; secondary ; 42A20
• 刊名：Statistics & Probability Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：120
• 期：Complete
• 页码：101-107
• 全文大小：362 K

文摘

We shall establish the law of large numbers for the discrete Fourier transform of random variables with finite first moment under condition 715216301821&_mathId=si1.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=ae3aff986fe984f6072744cedd17bfa3" title="Click to view the MathML source">P(|X_n|>x)≤P(|X₁|>x)$P(\left|X_n\right|>x)\le P(\left|X_1\right|>x)$ for all 715216301821&_mathId=si2.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=030e1aa73de35bb52b80db5a3d177984" title="Click to view the MathML source">x≥0$x\ge 0$; for 715216301821&_mathId=si3.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=54e7c795868048b43ba0c57ebe12f651" title="Click to view the MathML source">1<p<2$1<p<2$, we establish the Marcinkiewicz–Zygmund type rate of convergence for the discrete Fourier transform of random variables with finite 715216301821&_mathId=si4.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=4028bfdf049c82b7b38db90b344c8d2d" title="Click to view the MathML source">pth$pth$ moment under condition 715216301821&_mathId=si5.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=8d9602b340ea0a28a3ba051c9c10a92c">715216301821-si5.gif">$\frac{1}{n}{\sum }_{k=1}^{n}P(\left|X_k\right|>x)\le MP(\left|X_1\right|>x)$ for all 715216301821&_mathId=si2.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=030e1aa73de35bb52b80db5a3d177984" title="Click to view the MathML source">x≥0$x\ge 0$ and some positive constant 715216301821&_mathId=si7.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=5f9e905c1bae131b3824e542f57d2de9" title="Click to view the MathML source">M$M$.