<p>In the present paper, we consider the discrete Fourier transform <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S_n(t)=\sum _{k=1}^n e^{ikt}X_k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>e</mi> <mrow> <mi mathvariant="italic">ikt</mi> </mrow> </msup> <msub> <mi>X</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\{X_n, n\ge 1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a sequence of real valued random variables, and establish some limit behaviors for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(S_n(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which include the complete convergence, strong law of large numbers and moment convergence.</p>