Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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6 views

Is it true that $\sum_{n=1}^\infty \dfrac{Var(Y_n)}{n}<\infty$?

Suppose that $\{X_n\}$ is an i.i.d. sequence of random variables with $E|X_1|<\infty$.Define $Y_n=X_nI_{\{|X_n|<n\}}$ for all $n\geq1$. Is it true that $\sum_{n=1}^\infty ...
0
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0answers
6 views

Pareto distribution,fisher information, confidence interval

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, dont know how to do that on a keyboard. The rest is ...
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0answers
5 views

Density functions and estimators

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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1answer
11 views

Using characteristic functions to establish convergence

So I have found the Characteristic function of the variable $X_ \lambda$ to be: $$\psi_{X_\lambda}(t) = \psi_{b(\lambda)(Y_\lambda-\lambda)}(t)=\mathbf Ee^{itb(\lambda)(Y_\lambda-\lambda)}=\mathbf ...
1
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0answers
6 views

Martingale, increasing sequence of random variables

I want to prove the following: Let $\{X_n\} _{n \in \mathbb{N}}$ be a sequence of random variables such that $P( X_{n+1} \ge X_n)=1$. Then $$\{X_n\} _{n \in \mathbb{N}} \ \text{is a martingale} \ \iff ...
0
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0answers
9 views

Proof of the Uniform Law of Large Numbers

Theorem: Let $f:[0,1] \to \mathbb R$ be a measurable function bounded by $c$. Let $U_1,U_2,....\sim i.i.d~ \text{Uniform}(0,1)$. Then: $$ P \left( \lvert \frac{1}{n}\sum_{k=1}^nf(U_k)-\int_0^1f(x)dx ...
1
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0answers
9 views

Showing that a sequence of random variables has CLP.

This is an exercise that I am stuck at. I managed to solve (i) and (ii), which are relatively easy. Here the Feller condition in (i) is as below. Also, the central limit theorem I learned is like ...
1
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1answer
14 views

Show that $X_n/n$ does not converge almost surely

I am generally able to prove that a sequence of random variables $X_n$ converge almost surely to a random variable $X$ by using the following strategy: Take any typical sample point ...
3
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0answers
19 views

Interpretation of composite of random variable

Let $~f:[0,1] \to[0,\infty]$ be a measurable function bounded by $c \in \mathbb R$. Let $X_1,X_2,..,X_n \sim i.i.d ~\text{uniform}(0,1)$. How do I interpret the following statement: $$ Var(f \circ ...
2
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0answers
15 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
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1answer
38 views

Limit of random walk on $\mathbb{Z}$

$$\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}}=+\infty, \quad\liminf_{n \to \infty} \frac{S_n}{\sqrt{n}}=-\infty \quad P\text{-a.s.}$$ Here $S_n$ is a random walk on $\mathbb{Z}$. I managed to show ...
1
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1answer
16 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
1
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1answer
16 views

Convergence of random variables under different probability measures

I have a succession of random variables $X_n$ on $\Omega=[0,1]$ with $X_n=(1-\omega)^n$. I have to prove the convergence almost sure and/or in law in these case: $\mathbb P=\delta_{0}$ $\mathbb ...
3
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0answers
28 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
1
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0answers
6 views

multivariate convergence in distribution for Skorohod topology

If $X_n$ is a $R^d$-valued sequence of stochastic processes and $X$ is a $R^d$-valued the limiting process, all having Cadlag paths. I know what it means by $X_n \Rightarrow X$, i.e. $X_n$ converges ...
1
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1answer
24 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
1
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1answer
25 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...
-3
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1answer
37 views

How to find $E(X\mid X+Y=k)$ for binomial distribution? [on hold]

Let $X = \mathrm{Bin}(n,p)$ ; $Y = \mathrm{Bin} (m,p)$. How do I find $E(X\mid X+Y=k)$ for the binomial distribution?
3
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1answer
55 views

Random Variables and Moment Generating functions

Let $(X_i)_{i∈\Bbb{N^+}}$ be a sequence of i.i.d random variables and for $n ∈ \Bbb{N^+}$ set $S_n := \sum _{i=1}^{n} X_i$ and $Y_n := max(X_1, . . . , X_n)$. Assume that the moment generating ...
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2answers
18 views

CDF of a Uniform probability density function

I want to find Cumulative distribution function (CDF) of the following density function: $ f(x)= \begin{cases} 3/20 & \text{for } 2 \leq x \leq 4 \\[8pt] 4/20 & \text{for }4 < x \leq ...
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0answers
17 views

Are these random variables on the same probability space? [on hold]

In the following exercise in David Williams' Probability with Martingales, an argument is made that the random variables are under the same $\Omega$ and $\mathbb{P}$. It does not say so however in ...
0
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1answer
11 views

If the second moments are uniformly bounded, does $Y_n$ converge in $L^2$?

Let $\{X_n\}$ be a pairwise uncorrelated sequence of random variables such that there exists a fixed constant $c>0$ such that $E(X_n^2)\leq c$ for all $n\geq1$. Does it imply that for any ...
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0answers
23 views

If $X_n$ is Geo$(\lambda/n)$, does $X_n$ converge in distribution?

If $X_n$ is Geo$(\lambda/n)$, does $X_n$ converge in distribution? Clearly $\lambda>0$ and for all $n$, $\dfrac{\lambda}{n}\leq1$. Now, $$P(X_n\leq ...
2
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1answer
28 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
3
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1answer
30 views

probability question that just seems to easy to be the case

the game of mastermind starts in the following way: one player selects four pegs, each having six possible colors, places them in a line. the second player then tries to guess the sequence of colors. ...
0
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1answer
17 views

Probability, approximation by simple functions, boundedness and non-negativity.

In my probability class various results were announced requiring that a certain random variable was bounded. The specific example that interests me is the following: if $Y$ is $\mathcal{G}$-measurable ...
1
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0answers
24 views

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
5
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1answer
77 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|]$$ This question is a re-posting of An expectation inequality. I can ...
4
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2answers
49 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
-2
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0answers
41 views

limsup facts - which imply which? [on hold]

So I heard that for RVs $X_1, X_2, ...$ $\limsup X_n = \bigcap_{m \geq 1} \bigcup_{n \geq m} X_n$ $= \bigcap_{m \geq k} \bigcup_{n \geq m} X_n \forall k \in \mathbb{N}$ Does the last equality ...
0
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1answer
36 views

Show $\lim X_k < \infty$ is in tail sigma-algebra

Show $\lim X_k < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra. For ...
-1
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0answers
23 views

Show $\sum_k X_k < \infty$ is in tail sigma-algebra

Show $\sum_k X_k < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra. For ...
1
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0answers
17 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
-1
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1answer
40 views

Proposition on limsup

Given sets (or events) $A_1, A_2, A_3, ...$ and function $f: \mathbb{N} \to \mathbb{N}$, show that $\limsup A_{f(n)} \subseteq \limsup A_n \Leftrightarrow f(n) \to \infty$ This is what @Did told me ...
0
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0answers
17 views

Show $\limsup A_{2^n}$ is in the tail field.

Given events $A_1, A_2, A_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(A_n, A_{n+1}, ...)$ be their tail field. Without noting that $\limsup A_{2^n} \subseteq \limsup A_{n}$, prove $\limsup A_{2^n} ...
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1answer
15 views

Locally lipschitz implies zero quadratic variation? [on hold]

How can I prove that a locally Lipschitz function has zero quadratic variation? Thanks.
2
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1answer
53 views
+50

Sum of i.i.d. random variables and finding an upper bound

Problem: Suppose that $(X_i)_{i\in\mathbb{N}^+}$ is a sequence of i.i.d. random variables. For some $n\in\mathbb{N}^+$, let $S_n=\sum_{i=1}^n X_i$. Furthermore, let $a$ be a positive constant, and ...
3
votes
2answers
51 views

What is the intuition of why convergence in distribution does not imply convergence in probability

For me its very counter intuitive (that convergence in Probability and Distribution are not the same), because, conceptually, if two random variables have the same distribution, then they should be ...
2
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0answers
16 views

$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$

Let $c_i\in\mathbb R$, $a_i\geq0$ with $\sum_{i=1}^n a_i=1$, prove $$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$$ This inequality comes from there, when $X$ is ...
1
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0answers
25 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
2
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0answers
13 views

Good Problem Book for Convergence Concepts in Probability

I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are: Borel-Cantelli Lemmas Modes of Convergence and ...
2
votes
1answer
21 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
1
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1answer
23 views

Density function of minimum of random variables

Let $\{ X_i \}_{i=1, \dots,n }$ a set of i.i.d random variables whose density is defined by $f(\theta,x)=e^{-(x-\theta)}$ for $x>\theta$ and $f(\theta,x)=0$ for $x<\theta$. Where $\theta$ is a ...
0
votes
1answer
18 views

Example of an adapted but not progressively measurable process

I'm looking for an example of a stochastic process $X$ that is $\mathbb{F}$-adapted, but not progressively measurable. One example I found is the following: $(\Omega, \mathfrak{A}) = (\mathbb{R^+}, ...
3
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0answers
23 views

The copy-problem : Does any block of digits appear at least twice?

Suppose, $N$ random digits have been generated. Let $X$ be the largest natural number with the following property : There are natural numbers $i$ and $j$ with $i+X-1<j$ , such that the digits $i$ ...
0
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0answers
23 views

Maximizes the expected utility, Decisition making, statistics

QUESTION Can someone help me figuring out how to calculate this question, i just learned this stuff and i haven't do any example like this before... So i'm interpreted this question as: We need ...
1
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0answers
23 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
1
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0answers
10 views

On random functions taking values in the space of continous functions.

I upload the image of the passage that is unclear to me (Theoretical statistics by Keener): I do not understand how $W_1, W_2, \dots$ take values in $C(K)$. For every different realization of the ...
0
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0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
1
vote
1answer
16 views

Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration.

Let $\left(B_{t}\right)_{t\geq0}$ a Brownian motion. Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration. Remark: I try the following: It suffices to show ...