Cramér's decomposition theorem

<h2 id="the-precise-statement-of-the-theorem">The precise statement of the theorem</h2>
<p>Let a random variable ξ be normally distributed and admit a decomposition as a sum ξ=ξ1+ξ2 of two  independent random variables. Then the summands ξ1 and ξ2 are normally distributed as well.
</p><p>A proof of Cramér's decomposition theorem uses the theory of <a href="/facts/Entire_function/XNluKzu7">entire functions</a>.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Raikov%27s_theorem/JySmkXK2">Raikov's theorem</a>: Similar result for Poisson distribution.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lévy, Paul (1935). "Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées". J. Math. Pures Appl. 14: 347–402. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Cramer, Harald (1936). "Über eine Eigenschaft der normalen Verteilungsfunktion". Mathematische Zeitschrift. 41 (1): 405–414. doi:10.1007/BF01180430. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Linnik, Yu. V.; Ostrovskii, I. V. (1977). Decomposition of random variables and vectors. Providence, R. I.: Translations of Mathematical Monographs,  48. American Mathematical Society. <a href="/wiki/Yuri_Linnik" target="_blank">/wiki/Yuri_Linnik</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Cramér's decomposition theorem open-in-new

Cramér's decomposition theorem