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] Contenu 1 Énoncé du théorème 2 Preuve 2.1 Sufficiency of conditions ("si") 2.2 Necessity of conditions ("only if") 3 Exemple 4 Notes Statement of the theorem Let {style d'affichage (X_{n})_{nin mathbb {N} }} be independent random variables. The random series {textstyle sum _{n=1}^{infime }X_{n}} converges almost surely in {style d'affichage mathbb {R} } if the following conditions hold for some {displaystyle A>0} , and only if the following conditions hold for any {displaystyle A>0} : {somme de style d'affichage _{n=1}^{infime }mathbb {P} (|X_{n}|geq A)} converge. Laisser {style d'affichage Y_{n}=X_{n}mathbf {1} _{{|X_{n}|leq A}}} , alors {somme de style d'affichage _{n=1}^{infime }mathbb {E} [O_{n}]} , the series of expected values of {style d'affichage Y_{n}} , converge. {somme de style d'affichage _{n=1}^{infime }mathrm {var} (O_{n})} converge. Proof Sufficiency of conditions ("si") Condition (je) and Borel–Cantelli give that {style d'affichage X_{n}=Y_{n}} pour {displaystyle n} large, presque sûrement. Ainsi {displaystyle textstyle sum _{n=1}^{infime }X_{n}} converges if and only if {displaystyle textstyle sum _{n=1}^{infime }O_{n}} converge. Conditions (ii)-(iii) and Kolmogorov's Two-Series Theorem give the almost sure convergence of {displaystyle textstyle sum _{n=1}^{infime }O_{n}} .

Necessity of conditions ("only if") Supposer que {displaystyle textstyle sum _{n=1}^{infime }X_{n}} converges almost surely.

Without condition (je), by Borel–Cantelli there would exist some {displaystyle A>0} tel que {style d'affichage {|X_{n}|geq A}} for infinitely many {displaystyle n} , presque sûrement. But then the series would diverge. Par conséquent, we must have condition (je).

We see that condition (iii) implies condition (ii): Kolmogorov's two-series theorem along with condition (je) applied to the case {displaystyle A=1} gives the convergence of {displaystyle textstyle sum _{n=1}^{infime }(O_{n}-mathbb {E} [O_{n}])} . So given the convergence of {displaystyle textstyle sum _{n=1}^{infime }O_{n}} , Nous avons {displaystyle textstyle sum _{n=1}^{infime }mathbb {E} [O_{n}]} converge, so condition (ii) is implied.

Ainsi, 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 {displaystyle textstyle sum _{n=1}^{infime }Z_{n}=textstyle sum _{n=1}^{infime }(O_{n}-Y'_{n})} where for each {displaystyle n} , {style d'affichage Y_{n}} et {displaystyle Y'_{n}} are IID—that is, to employ the assumption that {style d'affichage mathbb {E} [O_{n}]=0} , puisque {style d'affichage Z_{n}} is a sequence of random variables bounded by 2, converging almost surely, et avec {style d'affichage mathrm {var} (Z_{n})=2mathrm {var} (O_{n})} . So we wish to check that if {displaystyle textstyle sum _{n=1}^{infime }Z_{n}} converge, alors {displaystyle textstyle sum _{n=1}^{infime }mathrm {var} (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 ( {style d'affichage 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: {somme de style d'affichage _{n=1}^{infime }pm {frac {1}{n}}.} Ici, " {displaystyle pm } " means that each term {displaystyle 1/n} is taken with a random sign that is either {style d'affichage 1} ou {style d'affichage -1} with respective probabilities {style d'affichage 1/2, 1/2} , and all random signs are chosen independently. Laisser {style d'affichage X_{n}} in the theorem denote a random variable that takes the values {displaystyle 1/n} et {displaystyle -1/n} with equal probabilities. With {displaystyle A=2} the summands of the first two series are identically zero and var(Yn)= {displaystyle n^{-2}} . The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely. D'autre part, the analogous series of (par exemple) square root reciprocals with random signs, à savoir {somme de style d'affichage _{n=1}^{infime }pm {frac {1}{sqrt {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, {somme de style d'affichage _{n=1}^{infime }(-1)^{n}/{sqrt {n}}} , which does converge.

Notes ^ Durrett, Rick. "Probabilité: Theory and Examples." Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Section 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", Université de Princeton. Presse (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