In many situations, it can be easy to apply Kolmogorov`s zero-one law to show that an event has a probability of 0 or 1, but surprisingly difficult to determine which of these two extremes is the right one. A general zero-one law was formulated by Kolmogorov (see [5]) as follows. Let $X_{1}, X_{2}dots$ a sequence of random variables, and let $f(X_{1}, X_{2},. ) $ be a borel measurable function, such that conditional probability In probability theory, Kolmogorov`s zero-one distribution, named after Andrei Nikolayevich Kolmogorov, states that a certain type of event, called a tail event, will almost certainly or almost certainly occur; That is, the probability of such an event occurring is zero or one. is an infinite sequence of independent random variables (not necessarily distributed identically). Let F {displaystyle {mathcal {F}}} be the algebra σ generated by X i {displaystyle X_{i}}. Then a tail event F ∈ F {displaystyle Fin {mathcal {F}}} is an event that is probabilistically independent of any finite subset of that random variable. (Note: F {displaystyle F}, which belongs to F {displaystyle {mathcal {F}}}, implies that membership in F {displaystyle F} is uniquely determined by the values of X i {displaystyle X_{i}}, but the latter condition is strictly weaker and insufficient to prove the zero-one distribution.) For example, the event that the sequence converges and the event that its sum converges are both final events. In an endless sequence of coin toss, a sequence of 100 consecutive heads that occur infinitely often is a tail event. for every $$N.
Under these conditions, the probability (*) is $0 or $1. For $X_{1} independent, X_{2}dots$, this leads to the zero-one law as stated at the beginning of the article. A more general statement of Kolmogorov`s zero-one law applies to sequences of σ independent algebras. Let (Ω,F,P) be a probability space and let Fn be a sequence of independent σ-algebras contained in F. Let be the smallest σ-algebra with Fn, Fn+1, .. Second, Kolmogorov`s zero-one law states that for every event, $1 is almost certainly $1 or $0 (depending on whether $f (X _ {1}, X _ {2},. ) is zero $ or not). This statement, in turn, follows from a sentence on Martingale (see [7], chap. III, section 1; Cap. VII, sect.
4, 5, 7 and commentaries; In Article 11, there is an analogue of the zero-one distribution for random processes with independent increments; This implies, in particular, that the sample distribution functions of a Gaussian process separable with a continuous correlation function are continuous at any point with a probability of $1$ or have a discontinuity of the second type with a probability of $1$ at each point; see also [8]). Let $Y_n = E(X| mathcal{F}_n)$. Then it`s $Y_n$ a martingale and $$sup_n E(| Y_n|) = sup_n E(| E(X| mathcal{F}_n)|) leq sup_n E(E(| X|| mathcal{F}_n)) = E(| X|) $$, where the obligation in the middle is due to Jensen`s conditional inequality. As P. Lévy proved in 1937 (see [6]), Kolmogorov`s theorem follows from a more general property of conditional probabilities, namely that you can find Doob`s convergence theorem in Williams Thm`s “probability with martingales”. 11.5. Your use of this feature and its translations is subject to all restrictions on use contained in the Project Euclid Website Terms of Use. The probability theory statement that any event (called an extreme event) whose occurrence is determined by elements arbitrarily distant from a sequence of independent random events or random variables has a probability of $0 or $1. This law extends to systems of random variables that depend on a continuous parameter (see below). Leave $mathcal{F}:={Ainmathcal{F}_{infty}: (1) text{ holds}}$, then we see that $mathcal{F}$ is a $lambda$ system: Could someone give me a proof of this theorem or reference? Thank you very much! You have requested the automatic translation of selected content in our databases. This feature is provided solely for your convenience and is in no way intended to replace human translation.
Neither Project Euclid nor the owners and publishers of the Content expressly make or disclaim any representations or warranties of any kind, express or implied, including, but not limited to, representations and warranties regarding the functionality of the translation function or the accuracy or completeness of the translations. Julius R. Blum. Pramod K. Pathak. “A Note on the Zero-One Law.” Ann. Math. Extra.
43 (3) 1008 – 1009, June 1972. doi.org/10.1214/aoms/1177692564. Now, heavy artillery exists according to Doob`s convergence theorem $Y_infty := lim_{n to infty} Y_n$ almost certainly. And since the sequence is dominated by $X$ (again conditional Jensen), we close the convergence $L^1$ and thus the convergence in probability. Thanks for A Blumenthal and Bunder, I was able to prove it. For the special case of a sequence $X _ {1} , X _ {2} dots $ of independent and identically distributed random variables has been shown (see [9]) that the probability is not only of any tail event, but also of an event invariant under each permutation of a finite number of terms of the sequence, is $0 or $1. Tail events associated with the analytical properties of sums of sequences of functions, e.g. power series with random terms, were also studied. Thus, the vague statement of Borel (1896) that, for “arbitrary coefficients”, the limit of the convergence disk is the natural limit of the analytic function represented by the coefficients, was put in the following precise form by H. Steinhaus [3]. Let $ X _ {1} , X _ {2} dots $ independent random variables evenly distributed over $ ( 0, 1 ) $ ( cf. Equal distribution), let $ a _ {k} $ with numbers and suppose the series power.
Remember that $mathcal{F}_{infty} = sigma left(cup_k mathcal{F}_k right)$. To show that $X_{infty} = E[X mid mathcal{F}_{infty}]$, let us invoke the determining property of conditional expectancy: We must show that $$ (*) ~~~~~~~E[X I_A ] = E[X_{infty} I_A] $$ for all $A in mathcal{F}_{infty}$. See for yourself that $(*)$ applies to all $A in mathcal{F}_k$ for every $k$ (note: use dominated convergence), and therefore $(*)$ applies to all $A incup_k mathcal{F}_k$, because it is an increasing union. Now, if $X_{1}, X_{2}dots$ is a sequence of independent random variables, then the probability that the series converges $sum_{k=} 1^infty X_{k}$ can only be $0 or $1. This fact (as well as a criterion for distinguishing these two cases) was established in 1928 by A.N. Kolmogorov (see [2], [5]). The statement of the distribution in the form of random variables is obtained from them by taking each Fn as the σ-algebra generated by the random variable Xn. A tail event is then by definition a measurable event with respect to the σ algebra generated by any Xn, but independent of a finite number of Xn. That is, a tail event is exactly one element of the intersection ⋂ n = 1 ∞ G n {displaystyle textstyle {bigcap _{n=1}^{infty }G_{n}}}.
Tail events are defined by infinite sequences of random variables. Suppose it has a radius of convergence $R > $0. Then the event (queue) that the $f$ function cannot be extended beyond the limit of disk $, | | leq R$ has a probability $1 $. B. Jessen [4] proved that any tail event associated with a sequence of independent random variables evenly distributed over $(0.1)$ has a probability of $0 or $1. Let $(Omega, mathcal{F},mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(mathcal{F}_k)_k$ be arbitrarily filtered and define $mathcal{F}_{infty}$ as the minimal $Ï$ algebra generated by $(mathcal{F}_k)_k$. Next, let`s first show that $X_{infty}$ is measurable with respect to $mathcal{F}_{infty}$.
For each $X_n$ is measurable in $sigma$-algebra $mathcal{F}_n$, i.e. in $mathcal{F}_{infty}$, then the limit a.s. $X_{infty}$ is measurable in $mathcal{F}_{infty}$. Thanks to the Dynkin set of systems, we have $mathcal{F}=mathcal{F}_{infty}$, so by definition we prove that $X_{infty}=E[X|mathcal{F}_{infty}]$. For individual tail events, the probability of being $0 or $1 was established in the early 20th century. So let $A_{1}, A_{2}dots$ a sequence of independent events. Let $A$ be the tail event where an infinite number of events $ A _ {k} $ occur, that is, intuitively the tail algebra $sigma$ contains events related to the behavior of the tail of the sequence $A_n,ninmathbb{N}$. I decided that this part was too long to comment on.
We know that with $X_k = E[X mid mathcal{F}_k]$ $$ X_k rightarrow X_{infty} $$ in $L^1$ and almost certainly for some $L^1$ we have random variables $X_{infty}$. arXivLabs is a framework that allows employees to develop and share new arXiv features directly on our website.