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

limsup facts - which imply which?

According to this answer on my previous question, $\limsup X_n = \bigcap_{n \geq 1} \bigcup_{m \geq n} X_n$ $= \bigcap_{n \geq 1} \bigcup_{m \geq n} X_n$ $= \bigcap_{n \geq k} \bigcup_{m \geq n} ...
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12 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 ...
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13 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 ...
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0answers
7 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 ...
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6 views

Show $\lim \frac{\sum_{j=1}^{k} X_j}{k} < \infty$ is in tail sigma-algebra

Show $\lim \frac{\sum_{j=1}^{k} X_j}{k} < \infty$ 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 ...
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1answer
18 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 ...
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14 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.
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21 views

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 ...
2
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1answer
19 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 ...
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21 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
12 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
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1answer
19 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 ...
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1answer
20 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 ...
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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^+}, ...
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0answers
22 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 $i$ ...
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0answers
21 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 ...
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0answers
19 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 ...
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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 ...
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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 ...
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1answer
15 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 ...
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1answer
27 views

probability of sequence of exactly 3 heads

A fair coin is tossed 5 times. What is the probability of getting a sequence of exactly 3 heads? Attempt: Sample space = $2^5 = 32$ \number of possible positions of first head = $5 - 3 + 1 = 3$ ...
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1answer
40 views

Class of graphs with symmetric random walk

Let $(V,E)$ be a graph and let $X_n$ be a random walk on the graph. At every step, the walker at $x$ jumps to one of the neighbors drawn uniformly at random among all the vertices $y$ such that there ...
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0answers
19 views

Find $v_k$ the probability of absorption at $N$ if the walk starts at $S_0=k$ for $0 \leq k \leq N$

Supose that $(S_n)_{n\geq0}$ is a random walk on $\{0,1,2,\dots,N\}$ with up prbability of $p$ and down probability of $(p-1)$. Find $v_k$ the probability of absorption at $N$ if the walk starts at ...
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1answer
23 views

Probability of winning a simple game

Consider two players, A and B start with 8 and 6 stones respectively. A rolls a six-sided die to determine how many stones to take from B. B performs the same task to determine how many stones to ...
3
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0answers
32 views

Computing MMSE and conditional expectation

Suppose we have three independent, zero mean, finite variance random variables $V,W,Z$ and where $W,Z$ are Gaussian random variables. These random variables form a new random variable $Y$ ...
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0answers
29 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \sim P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one efficiently ...
2
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1answer
39 views

choosing the right value to calculate a probability

A lot of $n$ itms contains $k$ defectives, and $m$ are selected randomly and inspected. How should the value of $m$ be chosen so that the probability that at least one defective item turns up is 0.90? ...
2
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0answers
7 views

Relationship between asymptotic full cardinality and full measure

Let $A$ be a finite set and for every integer $n \geq 1$ let $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#A^n} \to 1$. Now, let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on $A$. Is there a ...
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2answers
17 views

Expected Values of the product of a random variable and an indicator random variable

Let $X$ be a random variable $\in$ $L_{1}$ Given that $E[X]$ = $1$ , does that necessarily mean that : $E[X*1_{A}]$ = $P[A]$ ? My intuition is yes, since this is can be decomposed to $E[X]$ * ...
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1answer
14 views

Poisson counting process question, but correct answer not obtained by usual method

Ok here's the question: Fisherman Dan is out fishing by a stream. On average, 3 fishes per hour swim by but Fisherman Dan catches the fish with probability 1/2. It rains in average once per ...
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1answer
47 views

Prove absolute sum expectation

I have encountered the following problem, could someone provide me some hints on how to solve it? Assume that the sequence $(X_n)$ is i.i.d. with mean $0$ and variance $1$. For every $n\ge1$, let ...
3
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1answer
20 views

Independence of random variables and covariance in the limit.

Consider two sequences of random variables $(X_n)$ and $(Y_n)$ which converge in distribution to $X$ and $Y$ respectively, where $X$ and $Y$ are independent, but each pair $(X_n, Y_n)$ is not ...
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2answers
31 views

Split 1 in to n parts of size $2^{-k}$

I have the following problem: Let $n \geq 2$. Let $p_{i} = 2^{-k_{i}}, k_{i} \in \mathbb{N}$ s.t. $$p_{1} \geq p_{2} \geq ... \geq p_{n}$$ and $$\sum_{i=1}^{n} p_{i} = 1$$ I have to show the ...
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0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
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0answers
23 views

One dimensional Lazy random walk, $O(1/\sqrt{n})$?

Suppose that we have a Lazy 1-dimensional random walk $X_n$ valued in $\mathbb{Z}$, i.e. $$X_n = \sum_{i}^{n} \xi_i\;\;\;\;\;\;\;\;(\xi_i\;\text{iid}) $$ and $$\frac{1}{4}=P(\xi_1= 1)=P(\xi_1 =-1) ...
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1answer
17 views

Sub Sigma-Algebra and measurability

If a random variable $X$ is measurable with respect to a sub $\sigma$-algebra (let's say $\beta_{1}$), such that $\beta_{1}$ $\subset$ $\beta$ , is $X$ -necessarily- measurable with respect to the ...
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1answer
49 views

Infinite population mean?

When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
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0answers
19 views
+100

Mean value theorem for random variables

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$f(X+Y)=f(X)+f^\prime(X+\theta Y)(Y-X)$$ for real valued random variables $X$ and $Y$ and ...
2
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0answers
8 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
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2answers
19 views

Are as constant but not constant random variables trivial sigma-algebra-measurable? Converse?

Are almost surely constant random variables trivial sigma-algebra-measurable? These links suggest no: http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2004&task=show_msg&msg=1121.0001 ...
3
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1answer
22 views

Kurtosis of sum of Independent Random Variables

Suppose that $X$ and $Y$ are independent random variables with different expected values and variances. Suppose we define kurtosis as $$Kurt(X)=\frac{E[(X- \mu)^4]}{E[(X- \mu)^2]^2}$$ My question is ...
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2answers
34 views

Having two random generated natural numbers between 1 and 255, and generate out of it natural number between 1 and 256

Let's say you have two cube with 255 sides and you have to use them to simulate a single cube with 256 sides, how can I do it? $f(n)$ and $g(n)$ returns random number between 1 and 255. I thought ...
2
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0answers
31 views
+50

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
0
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1answer
24 views

Pearson coefficient and rate of change [on hold]

Given that two series $(x_1,....x_n)$ and $(y_1,....y_n)$ are linearly correlated How can I measure the change in the increment ($i\leq j$ ) $y_j−y_i$ as a function of $x_j−x_i$ , Pearson's r and ...
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0answers
32 views

How can two random variables are continuous infers that their jointly random variable is continuous?

We assume that $\forall a,b$ such that $a2+b2>0$, $aX+bY $ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Under which ...
3
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0answers
33 views

Why is the black-scholes model arbitrage free when $\sigma >0$?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
1
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1answer
11 views

Question regarding the density function of first n prediction

This is an example from Bertsekas' Introduction to Probability 2nd edition example 8.2 Consider now a variation involving the first $n$ dates. Assume that Juliet is late by random amounts $$X_1, ...
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0answers
26 views

A 'mix' of simple and lazy simple random walk

Consider a $\mathbb{Z}$ valued markov chain $X_n$ which evolves as follows. $$P(X_{n+1}=y | X_n) =\begin{cases} \frac{1}{2}, y=X_n+1, X_n-1, |X_n|>K \\ \frac{1}{4}, y = X_n-1 , y= X_n+1, ...
1
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2answers
36 views

Probability Runs

Paper is produced in a continuous process. Suppose that a brightness measurement is made on the paper once every half hour. We have fifteen measurements. The data was entered into R and the following ...