Probability of tail event using Kolmogorov’s 0-1 law

If $X_1,X_2,… $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I just don’t know how to prove it to be a tail event. Answer Observe that for each $n$, $\{X<\infty\} … Read more

Why isn’t D\mathcal D a sigma-algebra?

I came across the statement that if (Ω,F,P) is a probability space and E∈F then D:={A∈F∣A and E are independent} is a Dynkin system. I guess that D is not a sigma-algebra yet I can’t find a counterexample. Thus, we’d need a sequence (An) such that each An is independent from E yet their union isn’t. I’ve tried … Read more

Is there accepted notation for the pushforward measure that doesn’t mention P\mathbf{P}?

Let (Ω,F,P) denote a probability space, (S,M) denote a measurable space, and X:(Ω,F,P)→(S,M) denote a measurable function (thought of as a random variable). Then there is a pushforward measure induced on (S,M) (thought of as the probability distribution of X), which we could denote X∗(P), following Wikipedia. However, I like to imagine that X “knows” … Read more

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such examples is not at all obvious at first sight. The reason is that even though, in general, convergence in probability is weaker … Read more

If B(t)B(t) is Brownian motion then prove W(t)W(t) defined as follows is also Brownian motion

Let B(t) be standard Brownian motion on [0.1] Define W(t) as follows W(t)=B(t)−∫t0B(1)−B(s)1−sds Prove W(t) is also Brownian motion So I’m not sure how to deal with the integral here. In order to show it, too, is Brownian motion I think I would need to Make an argument that the transformation is linear and hence … Read more

Prove the convergence of a random variable

Given xn∼N(0,1n), is xn almost sure convergence or convergence in probability or convergence in distribution? How to prove it Answer By the Borel–Cantelli lemma, the sequence Xn∼N(0,n−1) converges to 0 almost surely as n→∞ if ∞∑n=1Pr for each \varepsilon>0. Using Markov’s inequality, we can obtain the following bound \Pr\{|X_n|>\varepsilon\}\le\frac{\operatorname E|X_n|^4}{\varepsilon^4}. Since X_n\sim N(0,n^{-1}), we know … Read more

Show E((X−Y)Y)=0E((X-Y)Y)=0

If EX2<∞ and E(X|G) is F-measurable then E(X|G)=EX There is one step in the proof which I don’t understand, set Y=E(X|G) and then why is E((X−Y)Y)=0, from a theorem I know that Y∈F is such that E(X−Y)2 is minimal Answer E[(X−Y)Y]=E[E((X−Y)Y|G)]=E[YE(X−Y|G)]=E[Y(E(X|G)−E(X|G))]=0. The first equality uses the Law of Iterated Expectations, the second “factoring out what … Read more

Example of using Delta Method

Let ˆp be the proportion of successes in n independent Bernoulli trials each having probability p of success. (a) Compute the expectation of ˆp(1−ˆp). (b) Compute the approximate mean and variance of ˆp(1−ˆp) using the Delta Method. For part (a), I can calculate the expectation of ˆp but got stuck on the expectation of ˆp2, … Read more

What is the probability that at least two of the nthn^{\rm th} biggest elements of A\mathbf{A} share the same column?

I have a random matrix A=[aij] for all i,j∈{1,…,n}. Every entry aij of the matrix A is generated randomly with exponential distribution. The aij are i.i.d and have the same parameter λ. Now, for each row i of A, I select the argument of the maximum element. That is, xi=argmax Let X_{ij} be the binary … Read more