# Tagged Questions

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### Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
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### What does tightness of convergence mean and why do we use this?

In a paper that I am reading it is written: We have $$(X(t_j))_{j=1}^k \xrightarrow{d} (Y(t_j))_{j=1}^k,$$ where $t_0=0, t_1 < \dotsb < t_k$. It further is written: In order to show that ...
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### Let $Y=\sup_n|X_n|$, Show that $Y<\infty$ a.s.

Suppose $\lim_{n\to\infty}X_n=X$ a.s. and $|X|<\infty$ a.s. Let $Y=\sup_n|X_n|$, Show that $Y<\infty$ a.s. If $\lim_{n\to\infty}X_n=X$ a.s. then $S_1:=\{w:\lim_{n\to\infty}X_n=X\}$ has ...
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### Convergence in probability of a random variable

I need to prove that $(X_n^2 -X)^2\to 0$ in probability $\Rightarrow X_n^2\to X$ in probability. I tried solving it with the triangle inequality, but it didn't get me anywhere. Is there another ...
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Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ... 1answer 39 views ### How to show convergence in distribution Let$([0,1],B,\lambda)$(B Borel Sigma-algebra) and$\lambda$the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ... 2answers 59 views ### Properties of a sequence of iid rv's I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c). 1answer 22 views ### Modes of Convergence of a particular random variable Let X_n \sim U([-1/n,1/n]) be uniform random variables on [-1/n,1/n] for n \in \mathbb{N}. Do the X_n converge, and if yes in what sense? I think it converges pointwise as for any x \in ... 1answer 34 views ### Convergence almost surely and B-C lemmas Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part. 0answers 28 views ### Asymptotic Bounds for the distribution of f_n(X_n). Let \{X_n\}_{n \in \mathbb{N}} be a sequence of \mathbb{R}^{k}-valued random variables defined on some probability space (\Omega, \mathcal{F}, \mathbb{P}) converging almost surely to X. ... 1answer 19 views ### Convergence in distribution problem I want to prove that, in (\mathbb{R},B(\mathbb{R})), we have that \frac{1}{n}\sum_{i=1}^{n}\delta_{\frac{i}{n}} converges to U_{[0,1]}. We need to prove, by definition, that \lim_{n \to ... 1answer 28 views ### Convergence in Probability and in Quadratic Mean for a sequence of random variables I have been trying to determine whether a sequence of random variables, X_1,X_2,\ldots,X_n, such that$$P\left(X_n= \frac{1}{n}\right)=1-\frac{1}{n^2}\\ \text{and}\\ ... 2answers 67 views ### Almost sure convergence of a sequence of random variables Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables$ (X_n) $, each with density function$g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ... 1answer 49 views ### Convergence in probability of iid normal random variables Let$X_1, X_2,\ldots$be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1)$X_n \to X_1$in law, (2)$X_n \not\to X_1$in ... 1answer 30 views ### Types of Convergence (Random Variables) Suppose that for every$n\ge 1$, the law of$X_n$is given by$P[X_n=n^2]=\beta_n$and$P[X_n=0]=1-\beta_n$, determine if$(X_n)_{n\ge 1}$converges in probability, in$L^1$or almost sure to zero, ... 0answers 35 views ### How do I prove the special case of the central limit theorem? Let$(X_n)$be an i.i.d. sequence such that$\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$and$\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$for some$\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ... 1answer 194 views ### Convergence of marginal distribtution Here I have a question which looks a little bit weird:$(q_n)_n$is sequence of probability density functions of the couple$(x,y) \in \mathbb R^2$,$p_n$is the marginal density of$q_n$, i.e. ... 1answer 27 views ### Convergence of random variable I've been facing the following problem: Let$(X_k, Y_k)_k$be a sequence of$2$-dimensional, independent random variables, each with uniform distribution over$ B(0,k) $Verify if the following ... 1answer 30 views ### tightness of sequence of degenerate probabilities If$\delta_x$denotes for$x\in \mathscr{R} $, the degenerate distribution at$x$, prove that the sequence$\delta_{x_n}$of probabilities on$(\mathscr{R,B})$is tight iff$x_n$is bounded. This is ... 1answer 25 views ### Existence of expected value with complex power Suppose that$X$be a random variable taking values on$(0,+\infty)$with density function$f(x)$and we have$\mathbb{E}(X^2)<\infty$. Can we conclude that $$\mathbb{E}(X^{-2-it}) ... 0answers 32 views ### problem on almost sure convergence Let {X_i} be iid with finite second moment. Let Y_n = \frac {2} {n(n+1)} \sum_{i=0}^n i*X_i , n\ge1 Show that Y_n \to E(X_1) I tried to define Z_i = \frac {2} {(n+1)} i*X_i Then Y_n = ... 1answer 48 views ### convergence in probability of function of random variables Suppose that X_1, X_2, \ldots, X_n be a sequence of i.i.d random variables. If we have E(|X_1|^k) <\infty for some k>0 and f(x) is a bounded continuous function on \mathbb{R}. Is the ... 0answers 39 views ### Markov Chain Geometric Convergence Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ... 1answer 61 views ### Convergence in probability implies convergence in mean under one additional condition Prove that if random variables X_n are dominated by an integrable random variable then E[X_n] \to E[X] follows if X_n converges to X in probability. Hint: Use the following theorem : A ... 2answers 155 views ### Convergence in probability to a non-measurable limit Let (\Omega, \mathcal{F}, P) be a probability space. Denote the Borel field on \mathbb{R} by \mathcal{B}. Let \mu: \Omega \rightarrow [0,\infty) be a not-necessarily-measurable function and, ... 1answer 42 views ### Convergence in probability of sample variance X_n s are a sequence off iid random variables with E(X_n) = \mu, V(X_n)= \sigma$$^2$ and $\bar X = \sum$ $\frac{X_i}{n}$. Then show that $\frac1n$ $\sum (X_i - \bar X )^2\to\sigma^2$ in ...
Let $f_n$ be $2^n$ times the indicator of the set of $x$ in the unit interval for which the digit from $n+1$ to $2n$ is zero in the dyadic expansion of $x$ (lets call it $A_n$). I have to show that ...
Suppose $X_1, X_2,\dots,$ be an independent sequence of random variables and $E[X_n] = 0 \forall n$ and $\sum_{n=1}^{\infty} \operatorname{Var}(X_n) < \infty$. I need to prove that ...