We shall establish the law of large numbers for the discrete Fourier transform of random variables with finite first moment under condition class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si1.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=ae3aff986fe984f6072744cedd17bfa3" title="Click to view the MathML source">P(|Xn|>x)≤P(|X1|>x)class="mathContainer hidden">class="mathCode"> for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si2.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=030e1aa73de35bb52b80db5a3d177984" title="Click to view the MathML source">x≥0class="mathContainer hidden">class="mathCode">; for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si3.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=54e7c795868048b43ba0c57ebe12f651" title="Click to view the MathML source">1<p<2class="mathContainer hidden">class="mathCode">, we establish the Marcinkiewicz–Zygmund type rate of convergence for the discrete Fourier transform of random variables with finite class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si4.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=4028bfdf049c82b7b38db90b344c8d2d" title="Click to view the MathML source">pthclass="mathContainer hidden">class="mathCode"> moment under condition class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si5.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=8d9602b340ea0a28a3ba051c9c10a92c">class="imgLazyJSB inlineImage" height="21" width="231" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301821-si5.gif">class="mathContainer hidden">class="mathCode"> for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si2.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=030e1aa73de35bb52b80db5a3d177984" title="Click to view the MathML source">x≥0class="mathContainer hidden">class="mathCode"> and some positive constant class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301821&_mathId=si7.gif&_user=111111111&_pii=S0167715216301821&_rdoc=1&_issn=01677152&md5=5f9e905c1bae131b3824e542f57d2de9" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">.