Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...
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16 views
A fact about independent exponentially distributed RVs
Let's say $Y_i$ are independent exponentially distributed with rates $c_i$ which can be assumed to be all in $(0, \infty)$ but not necessarily the same. Whenever $t\geq0, K\geq0$ let ...
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0answers
28 views
Is it possible to have $\mathbb E S_n \to \infty$ but $\inf S_n = -\infty$ a.s.?
The original question is:
Let $X_1, X_2, \ldots$ be i.i.d. with $\mathbb P (X_1 = 1) = p > 1/2$ and $\mathbb P (X_1 = −1) = 1 − p$, and let $S_n = X_1 + \cdots + X_n$. Let $\alpha = \inf\{m : ...
1
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0answers
27 views
Convergence in probability of a specific sum
Consider autoregression AR(1):
$$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$
$\{\varepsilon_t\}$ - i.i.d. random variables with $E\varepsilon_1 = 0,$ $E\varepsilon_1^2<\infty$
...
1
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2answers
27 views
Random Variables from [0,1] - Integration Limits
I was wondering if someone could help me understand the first steps I should take for solving the next problem:
Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with ...
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1answer
28 views
the distribution of the inverse of a standardized uniform variable
If $u$ is a standardized uniform variable, what is the mean and variance of $x=\frac{1}{u}$? What can be said about the distribution of $x$?
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1answer
26 views
Sequences of i.i.d. subgaussian RVs and uniform integrability
Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)?
Intuitively it appears to be so; if we take for example $\{a_j\}$ ...
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0answers
29 views
Total set of functions in $L^2(\Omega)$
Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
2
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2answers
39 views
Laplace transform converging to zero
I have a sequence $X_n$ of random variables, whose Laplace transform $F_n(\lambda)=\mathbb{E}(e^{-\lambda X_n})$ satisfy for every $\lambda\geq0$
$$F_n(\lambda)\to1\qquad (n\to\infty).$$
Is it enough ...
3
votes
3answers
85 views
Central Limit Theorem Definition
My friend and I have a bet going about the definition of the Central Limit Theorem.
If we define an example as a number drawn at random from some probability density function where the function has a ...
1
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1answer
31 views
Counting probability question-what is the sample space in this problem?
Hi folks this is a self learn probability (counting) question from DeGroot. The question is:
Suppose that a box contains r red balls and w white balls. Suppose also that the balls are drawn out ...
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1answer
34 views
Sum of positive i.i.d. random variables.
Let $X_1, X_2, \ldots, $ be i.i.d. random variables with $X_1 > 0$. Let $S_n = \sum_{m=1}^n X_m$. Can we conclude $[\sup_{n \ge 1} = \infty$] almost surely?
Assuming the statement is true, by ...
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1answer
24 views
Continuous mapping theorem for convergence in $L^2$
This question is related to Analogue of continuous mapping theorem for convergence in $L^2$ but has narrower conditions.
Is it true that:
If
1) $g$- continuous $\mathbb{R}\to\mathbb{R}$ function,
...
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0answers
47 views
Central Limit Theorem: probability density function of the true mean?
A friend and I are arguing over the meaning of the Central Limit Theorem. I am stating that the normal distribution seen by taking the mean of a large number of samples is a probability density ...
3
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0answers
50 views
A problem on probability theory and geometry
I am stuck on the following problem: given a circle of radius $R$ I put randomly $N$ points inside the circle. What is the probability to have the distances of every point from the other greather than ...
3
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0answers
39 views
Intuition for the optimality of bold play
There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
2
votes
1answer
26 views
Truncated expectation = 0 implies function vanishes a.e.
Suppose $X$ is a positive random variable and $ A \subset \Omega$ such that $\mathbb{E}(X1_{A}) = 0$. Show $ X = 0 $ a.e. on $A$.
This basic problem has really stump me. If we define the event $A' = ...
1
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0answers
47 views
Borel-Cantelli (?) to show that a lim sup inequality of rapid decay random variables holds with prob. 1
A random variable $\xi$ in probability space $(\Omega,\mathcal{A},P)$ is said to have $c$-rapid decay if $P(\xi > k)\leq e^{-ck}$ for sufficiently large $k$. I'm given a sequence $\xi_n$ of random ...
2
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1answer
50 views
Analogue of continuous mapping theorem for convergence in $L^2$
Could you help please:
Is there any analogue of continuous mapping theorem for convergence of sequence of random variables in $L^2$?
I mean:
If $g$ is a continuous function $\mathbb{R}\to\mathbb{R}$ ...
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votes
0answers
17 views
From Q matrix to Markov Chain
We are in the setting of a continuous time MC, as defined by Liggett in his book on continuous time markov processes, on a countable state space $S$. All of his MCs are defined on the space of right ...
2
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1answer
23 views
Convergence of measures concentrated on graphs of functions
Suppose we have a sequnece of probability measures $\{ \mu_n\}$ on $\mathbb{R}^{d+m}$ where each element is concentrated on the graph of a function, that is $\mu_n$ is concentrated on $\{ (x,f_n(x)) ...
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1answer
20 views
Is that Probability function only for discrete case?
Most of the books and sites define Probability function for discrete case that is they use the term as the synonym of Probability mass function.
Is that Probability function define for only ...
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0answers
24 views
a question about transformation from an article
I have a question from a proof in an article:
http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/MultipleComparision/Simes86pdf.pdf
I was reading the above article and tried to ...
1
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2answers
41 views
Does $X_n \stackrel{Prob}{\longrightarrow} X$, $X \in L^2$ imply $X_n \stackrel{L^2}{\longrightarrow} X$?
Let $X_n$, $n\in \mathbb{N}$ be a sequence of random variables which converges in probability to $X$, i.e. $X_n \stackrel{Prob}{\longrightarrow} X$. Furthermore it is known that $X \in L^2$. Does this ...
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0answers
39 views
Will the sum tend to zero in probability?
Consider autoregression AR(1):
$$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$
Let $\{u_t^{0}\}$ be the stationary solution of this equation when $\beta=\beta_0$ and $\{u_t\}$ - ...
1
vote
1answer
25 views
Process with Feller stochastic kernel?
Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow [0,1]$ be a probability measure.
Let $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ be measurable, and continuous in the first argument.
...
4
votes
0answers
59 views
A case of the central limit theorem
I want to show that
$$\frac{\sum_{k=1}^N X_k}{\sqrt{\sum_{k=1}^N X_k^2}} \overset{N\to\infty}{\to} \mathcal{N}(0,1)\text{ in distribution,}$$
where $X_1,X_2,\ldots$ is a sequence of iid random ...
2
votes
1answer
27 views
Does p(a,b,c)=p(a,b)p(a,c) hold when b and c are independent?
I am reading through a thesis, it states that given $b$ and $c$ being independent:
\begin{equation}
p(a\mid b,c) := p(a\mid b) p(a\mid c)
\end{equation}
This would imply just using the definition of ...
1
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2answers
49 views
Convergence of $\int_{A_n} f$ to $0$
I am looking for a name or a reference in a textbook for the following result in order to quote it.
For any $f\in L^1(\mathbb{R})$-integrable function, we have
$$\lim_{n\to\infty}\int_{A_n} ...
5
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0answers
52 views
Azuma's inequality to McDiarmid's inequality?
I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
1
vote
1answer
38 views
Limit of characteristic functions
Let $\xi_1 ... , \xi_n$ be iid with $E \xi_i^2 < \infty $
what is $$ \lim _{n\rightarrow \infty} \varphi_{\bar{\xi}}$$
where $\varphi$ is the caracteristic function and $\bar{\xi}$ the mean of all ...
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0answers
44 views
No subgraph with average degree greater than $3c$ for $G(n,m)$ where $m=cn/2$
Consider the graph $G_{n,m}$, which has $n$ nodes and $m$ random edges. Let $m=cn/2$ where $c>0$ is a constant. I want to prove that with high probability (i.e. with probability $1-o(1)$), there is ...
5
votes
1answer
43 views
Laplace transform of a random variable
My professor says that the Laplace transform of a nonnegative RV uniquely determines the RV up to distributional equality among all nonnegative RVs. He says one can argue this by appealing to a fact ...
1
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0answers
19 views
Relation between conditional variances
Let $X,A,Y_1,\dots,Y_n$ be random variables defined on some probability space $(\Omega,\mathcal{F},P)$. Suppose that $Y_1,\dots,Y_n$ are conditionally independent and identically distributed given the ...
2
votes
0answers
36 views
Strictly monotone probability measure
Let $m$ be a probability measure on $X \subseteq \mathbb{R}^n$.
Let $f: X \rightarrow \mathbb{R}$ be measurable.
Assume that there exists $\epsilon > 0$ such $m\left( \{ x \in X \mid |f'(x)| \leq ...
2
votes
0answers
31 views
Criteria for $L^1$ convergence looking at Laplace transforms
Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
1
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1answer
19 views
How to compute conditonal probability on multiple conditions on the right side?
Is there a formula for a conditional probability of many "conditions" I.e:
$P(X|Y,Z,\ldots)$ That is how to compute $P(X)$ given $X,Y,\ldots$ Or given separately $P(X|Y)$ and $P(X|Z)$ is there a ...
1
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0answers
64 views
Likelihood Function for the Uniform Density.
Let the random variable $X$ have a uniform density given by
$$
f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]}
$$
where $-\infty\leq\theta\leq\infty $
the likelihood function for a sample of ...
0
votes
0answers
17 views
Help with an asymptotic proof? [duplicate]
I am having difficulty with a proof of an asymptotic result in maximum likelihood estimation theory. I don't believe any statistical theory however is needed to solve this problem, rather I think ...
1
vote
1answer
23 views
How to choose two random variables taking values in a finite space, with given distributions, such that probability that they are equal is maximized?
Let A = {1, 2, ..., n}, and let X and Y be two random variables on the same space, taking values in A, and distributions given by:
P(X = i) = $p_i$, and P(Y = i) = $q_i$, for any i in {1,2,...,n}
...
2
votes
1answer
59 views
Likelihood Functon.
$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ .
What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ ...
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0answers
52 views
Help with an asymptotic proof?
I am having difficulty with a proof of an asymptotic result in maximum likelihood estimation theory. I don't believe any statistical theory however is needed to solve this problem, rather I think ...
4
votes
0answers
28 views
Intuition on continuty in probability/mean square of a process
How to explain that a process is continuous in probability? I know the definition, but what does it mean? The same with continuity in mean square.
2
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0answers
19 views
When is a Gaussian field almost surely bounded?
Let $p\ge1$ and assume given a (deterministic) function
$$
R:\mathbb{R}^p\times\mathbb{R}^p\to\mathbb{R}
$$
such that $R$ is symmetric, meaning that $R(x,y)=R(y,x)$ for all $x,y\in\mathbb{R}^p$, and ...
1
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0answers
32 views
Inequality of covariances between a bivariate normal vector and its indicator functions
Why holds for a standardized bivariate normal vector $Z:=(Z_1,Z_2)$ that
\begin{equation}
|\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|?
\end{equation}
...
-1
votes
0answers
28 views
Proof in Probability Distribution Functions
The number of heads a coin comes up tails when tossed n times is denoted by random
variable X. Suppose that for each toss, the coin will appear heads with probability z.
(a) The probability mass ...
3
votes
0answers
23 views
Converge in Distribution imply expectation convergence
Let $X_n$ and $X$ random variables in not the same probability space and let $X_n \to X$ in distribution. If $f(x)\leq C(1+x^m)$, for some $C,m>0$ constants, then I have to prove that ...
2
votes
4answers
54 views
Is this Expectation finite?
How do I prove that
$\int_{0}^{+\infty}\text{exp}(-x)\cdot\text{log}(1+\frac{1}{x})dx$
is finite? (if it is)
I tried through simulation and it seems finite for large intervals. But I don't know ...
0
votes
1answer
28 views
understanding probability distribution notation
Assume that $\mu$ is a probability distribution on $[n]$, let $A\subseteq[n] $ be a probability event. What does it mean : $Pr_\mu[A]$ ?
0
votes
1answer
31 views
Martingale and submartingale
When $X_n$ is a martingale we know $(X_n)^+$, i.e. the positive part of $X_n$, is also a submartingale.
Is this true? It is easy to show that this is SUB-m.g. but I couldn't find a counter-example for ...
0
votes
2answers
39 views
Compare Expected Value of two function
I have a problem in comparing expected value of two functions.
Let $X,Y$ be two i.i.d. random variables, exponentially distributed with mean $\lambda$.
I want to claim that $E [\log(1+ ...
