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. I just went through a book that proves many interesting and rather difficult results on Brownian motion (PDF link, website link), and it seems that Kolmogorov`s zero-one law applies to most of them. If you create a „random network“ with some probability p of edges between nodes (see the article above for exact definitions), then there exists an infinite cluster with a probability of zero or one. But for a given value of p, it cannot be trivial to determine which one. Let be the smallest algebra σ with Fn, Fn+1, .. Second, Kolmogorov`s zero-one law states that for each event 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. The Rademacher–Paley–Zygmund theorem answers this question. A discussion (no proof!) of this theorem and related problems is given here: Using Kolmogorov`s zero-one distribution, it can be shown that Pr{AX = BX} is 0 or 1, where AX is the class of problems that can be solved by complexity class A with oracle access to the X language. X is chosen uniformly in all languages.
(The set of all languages is basically the set of powers of Z, so one can think of this set as [0,1] and then reformulate the probabilistic statement as a statement about the measure of the set to make it more accurate.) 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. The first question posed in percolation theory is whether there is an infinite open cluster. The zero-one law is true because, as David Speyer said above, the existence of an infinite cluster is invariant under finite changes of edges. Equivalently, the existence of an infinite cluster is a translational invariant event. Thus, this probability is zero or one, but depends on p, the parameter of the system (the probability that a given link is open). The Nice theorem is that there is a critical pc parameter that depends only on the structure of the network. Using Fourier transforms, a standard Brownian motion Xt in the interval 0≤t≤1 can be decomposed as $$ X_t = At + sum_{n=1}^inftyfrac{1}{sqrt{2}pi n}left(B_n(cos 2pi nt – 1)+C_nsin 2pi ntright) $$, where A, Bn, Cn are independent normals with an mean of 0 and variance 1. It follows that any property of Brownian motion that is unchanged with the addition of a linear combination of sine, cosine, and linear terms is a tail event and has a probability of zero or one according to Kolmogorov`s zero-one distribution. We know that Brownian motion is nowhere differentiable (with probability 1).
In probability theory, Kolmogorov`s zero-one law, 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. Here is the proof of Kolmogorov`s zero-one distribution and the lemmas used to prove it in Williams` probability book: Certainly, if p < pc, then θ(p) = 0, since there is no open cluster to which 0 belongs. For other values of p, θ(p) does not need to be equal to one, because the zero-one distribution does not apply: you can truncate 0 of an infinite cluster by a finite number of changes (which close the bonds around 0); Similarly, the event is not translationally invariant. I`m sure there must be situations where the answer is harder to determine, but the example I was interested in about the 0-1 law (which I took from an excellent lecture by W. Russell Mann about ten years ago) is this: Suppose $A in mathfrak{T}$. Then A is in the model of. I don`t know. Help please? According to my above decomposition of Brownian motion, all these definitions and statements refer to tail events and we know that they must have a probability of 0 or 1 of being true.
In fact, slow and fast time sets are defined in terms of tail events, so any measurable statement about these sets must always be true or always false with probability one, and all their measurable functions, such as their fractal dimensions, must be deterministic constants with probability one, even if it is difficult to calculate, what they are. The same goes for many other properties of Brownian motion in the book I linked to – these are tail events and therefore always true or false with probability. There are a number of good examples of percolation theory: en.wikipedia.org/wiki/Percolation_theory Intuitively, I understand that. I`m just wondering how I can prove it rigorously. Although we know the answer to this question for some groups of complexity classes such as P and NP (Pr{PX = NPX}=0), I am sure there are many complexity classes for which we do not know the answer. It becomes more interesting when you look at the modulus of continuity of Brownian motion. For each time t, the law of the iterated logarithm says that $$limsup_{hdownarrow 0}frac{| X_{t+h}-X_t|} {sqrt{2hloglog (1/h)}}=1 $$ with probability 1. From this, it can be said that Brownian motion encounters this limit almost everywhere (but not everywhere – there are extraordinary moments). More generally, they show that the probability we always hold the following values: $$ limsup_{hdownarrow 0}frac{| X_{t+h}-X_t|} {sqrt{2hlog(1/h)}}le 1, limsup_{hdownarrow 0}frac{| X_{t+h}-X_t|} {sqrt{h}}GE 1. $$ With probability one, these limits will be reached. The moments when left inequality is equality are fast times and slow times are those when law is equality.
In the book I linked, they calculate a lot of things about these fast and slow times, like their fractal dimensions. So $Ain mathfrak{X}_n$ for some $n$ and since $mathfrak{X}_n$ and $mathfrak{T}$ are independent $sigma$ algebras, we can conclude that $P(Acap B)=P(A)times P(B)$. A transformation preserving an invertible measure on a standard probability space that obeys the 0-1 distribution is called Kolmogorov automorphism. [clarification needed] All Bernulli automorphisms are Kolmogorov automorphisms, but not vice versa. The existence of an infinite cluster in the context of percolation theory also obeys the 0-1 law.
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