Kolmogorov's three-series theorem

Kolmogorov's three-series theorem In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions. Kolmogorov's three-series theorem, combined with Kronecker's lemma, can be used to give a relatively easy proof of the Strong Law of Large Numbers.[1] Conteúdo 1 Declaração do teorema 2 Prova 2.1 Sufficiency of conditions ("E se") 2.2 Necessity of conditions ("only if") 3 Exemplo 4 Notes Statement of the theorem Let {estilo de exibição (X_{n})_{nin mathbb {N} }} be independent random variables. The random series {textstyle sum _{n=1}^{infty }X_{n}} converges almost surely in {estilo de exibição mathbb {R} } if the following conditions hold for some {displaystyle A>0} , and only if the following conditions hold for any {displaystyle A>0} : {soma de estilo de exibição _{n=1}^{infty }mathbb {P} (|X_{n}|geq A)} converge. Deixar {estilo de exibição Y_{n}=X_{n}mathbf {1} _{{|X_{n}|leq A}}} , então {soma de estilo de exibição _{n=1}^{infty }mathbb {E} [Y_{n}]} , the series of expected values of {estilo de exibição Y_{n}} , converge. {soma de estilo de exibição _{n=1}^{infty }matemática {era} (Y_{n})} converge. Proof Sufficiency of conditions ("E se") Condition (eu) and Borel–Cantelli give that {estilo de exibição X_{n}=Y_{n}} por {estilo de exibição m} large, quase com certeza. Por isso {soma de estilo de texto de estilo de exibição _{n=1}^{infty }X_{n}} converges if and only if {soma de estilo de texto de estilo de exibição _{n=1}^{infty }Y_{n}} converge. Conditions (ii)-(iii) and Kolmogorov's Two-Series Theorem give the almost sure convergence of {soma de estilo de texto de estilo de exibição _{n=1}^{infty }Y_{n}} .

Necessity of conditions ("only if") Suponha que {soma de estilo de texto de estilo de exibição _{n=1}^{infty }X_{n}} converges almost surely.

Without condition (eu), by Borel–Cantelli there would exist some {displaystyle A>0} de tal modo que {estilo de exibição {|X_{n}|geq A}} for infinitely many {estilo de exibição m} , quase com certeza. But then the series would diverge. Portanto, we must have condition (eu).

We see that condition (iii) implies condition (ii): Kolmogorov's two-series theorem along with condition (eu) applied to the case {displaystyle A=1} gives the convergence of {soma de estilo de texto de estilo de exibição _{n=1}^{infty }(Y_{n}-mathbb {E} [Y_{n}])} . So given the convergence of {soma de estilo de texto de estilo de exibição _{n=1}^{infty }Y_{n}} , temos {soma de estilo de texto de estilo de exibição _{n=1}^{infty }mathbb {E} [Y_{n}]} converge, so condition (ii) is implied.

Desta forma, it only remains to demonstrate the necessity of condition (iii), and we will have obtained the full result. It is equivalent to check condition (iii) for the series {soma de estilo de texto de estilo de exibição _{n=1}^{infty }Z_{n}=textstyle sum _{n=1}^{infty }(Y_{n}-Y'_{n})} where for each {estilo de exibição m} , {estilo de exibição Y_{n}} e {displaystyle Y'_{n}} are IID—that is, to employ the assumption that {estilo de exibição mathbb {E} [Y_{n}]=0} , desde {estilo de exibição Z_{n}} is a sequence of random variables bounded by 2, converging almost surely, e com {matemática de estilo de exibição {era} (Z_{n})=2mathrm {era} (Y_{n})} . So we wish to check that if {soma de estilo de texto de estilo de exibição _{n=1}^{infty }Z_{n}} converge, então {soma de estilo de texto de estilo de exibição _{n=1}^{infty }matemática {era} (Z_{n})} converges as well. This is a special case of a more general result from martingale theory with summands equal to the increments of a martingale sequence and the same conditions ( {estilo de exibição mathbb {E} [Z_{n}]=0} ; the series of the variances is converging; and the summands are bounded).[2][3][4] Example As an illustration of the theorem, consider the example of the harmonic series with random signs: {soma de estilo de exibição _{n=1}^{infty }PM {fratura {1}{n}}.} Aqui, " {estilo de exibição pm } " means that each term {estilo de exibição 1/n} is taken with a random sign that is either {estilo de exibição 1} ou {estilo de exibição -1} with respective probabilities {estilo de exibição 1/2, 1/2} , and all random signs are chosen independently. Deixar {estilo de exibição X_{n}} in the theorem denote a random variable that takes the values {estilo de exibição 1/n} e {displaystyle -1/n} with equal probabilities. With {displaystyle A=2} the summands of the first two series are identically zero and var(Yn)= {estilo de exibição n^{-2}} . The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely. Por outro lado, the analogous series of (por exemplo) square root reciprocals with random signs, nomeadamente {soma de estilo de exibição _{n=1}^{infty }PM {fratura {1}{quadrado {n}}},} diverges almost surely, since condition (3) in the theorem is not satisfied for any A. Note that this is different from the behavior of the analogous series with alternating signs, {soma de estilo de exibição _{n=1}^{infty }(-1)^{n}/{quadrado {n}}} , which does converge.

Notes ^ Durrett, Rick. "Probabilidade: Theory and Examples." Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Seção 1.8, pp. 60–69. ^ Sun, Rongfeng. Lecture notes. http://www.math.nus.edu.sg/~matsr/ProbI/Lecture4.pdf Archived 2018-04-17 at the Wayback Machine ^ M. Loève, "Probability theory", Princeton Univ. Imprensa (1963) pp. Sect. 16.3 ^ W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) pp. Sect. IX.9 Categories: Mathematical seriesProbability theorems