# Tagged Questions

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### Corollary of Kolmogorov zero-one law

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
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### Notation in sequences of random variables

I read in statistics books that a sequence of random variables are often written as ${X_n}$. But in all the theorems it just says $X_n$. Why is that? And does ${X_n}$ symbolize the WHOLE sequence ...
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### Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely

I'm trying to solve the following Problem: Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there ...
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### Does this sum converge or diverge?

Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal ...
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Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$\hat ... 0answers 60 views ### weak convergence and composition Assume X_n is a sequence of random variables defined on a common probability space and X_n converges weakly (in distribution) to X as n \to \infty. Assume u_n is a sequence of integer valued ... 0answers 69 views ### A nice sequence of random variables Let f:U\mapsto \mathbb{R}^k with U\subset \mathbb{R} be a smooth injective function. Suppose that \sqrt n(Y_n- Y)\to N(0,\Omega) in distribution with Y=f(X). Define X_n by ... 0answers 34 views ### Robbins-Siegmund like theorem for a nonlinear system Recently I came to know about this theorem due to Robbins and Siegmund which states the following: Let us have on a probability space (\Omega, \mathcal{F},P), a filtration \{\mathcal{F}_n\} and ... 1answer 69 views ### Is there a sequence of i.i.d. random variables that is eventually monotonically decreasing? Here is the problem I'm struggling with: Let (X_n) be is a sequence of independent and identically distributed random variables. What is the probability that the sequence is monotonically ... 1answer 92 views ### Three series of Kolmogorov Let X_n\geqslant 0 be a sequence of independent random variables. The following are equivalent: i) \sum_{n=1}^{\infty}{ X_n} <\infty a.s ii) \sum_{n=1}^{\infty}{ \mathbb P(X_n>1)} ... 0answers 63 views ### Given an innovations sequence, calculate transformation matrix, A Let Y_1,Y_2,Y_3,X)^T be a zero mean random vector with correlation matrix,$$ \begin{pmatrix} 2 & 1 & 1 & 2 \\ 1 & 2 & 1 & 2 \\ 1 & 1 & 2 & 2 \\ 2 & 2 & 2 ...
Let $X_0, X_1$ and $\{a_n,n\geq0\}\sim$Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1$$ Then it follows that ...