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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178 views

What are some characterizations of the strong and total variation topologies on measures?

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for ...
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66 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
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70 views

Question regard almost sure convergence

I need an example of an i.i.d. sequence of random variables $(X_i)$ such that $$ X_1\cdot\left(\frac{1}{n}\sum_{i=1}^nX_i\right)\not\to X_1\mathbb{E}(X_1) $$ almost surely as $n\to\infty$. ...
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115 views

Upper semicontinuity of a probability measure

Let $m$ be an atomless probability measure on $\mathbb{R}^m$. Consider $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that for all $v \in \mathbb{R}^m$, $x \mapsto f(x,v)$ is ...
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105 views

Mathematics Courses for an Economist

I am an Economist and I am interested in further developing my mathematical knowledge and skills. I would like to get your opinions on the topics that I should cover and which are also important for ...
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142 views

Example of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?

Let $$dX_t = a(t,X_t) \, dt + b(t, X_t) \, dW_t, \quad t \in [0,T]$$ be a stochastic differential equation, where $W$ is an $m$-dimensional Brownian motion, $X_0 = x \in \mathbb{R}^d$, and the ...
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107 views

Bounding the maximum of the sum of squared iid. normal variables

I am trying to work my way through a proof and I am stuck on the following problem: Assume for $i = 1, \dots, p$ that $x_i \sim N(0,\sigma^2)$. For a binary $p$-dimensional vector $\gamma$ define: ...
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241 views

On computing a conditional expectation for countable-co-countable sigma-algebras

Let $S=[0,1]$ equipped with its Borel $\sigma$-algebra ${\cal B}. $ Assign on it a purely atomic probability measure, $P $. Let ${\cal A} $ be the sub-$\sigma$-algebra generated by the countable and ...
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33 views

Weak convergence of a collection of random variables starting from a random index

Imagine for any $k \in \mathbb{N}$ the following holds: $$ (X_{r}^{(1)}, X_{r}^{(2)} \dots X_{r}^{(k)}) \Rightarrow (Y_1, Y_2, \dots, Y_k), $$ where $Y_1, Y_2, \dots, Y_k$ are i.i.d. random variables. ...
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77 views

trace class norms of random matrices

We denote by $||.||_1$ the trace class norm. on $M_n$.Let $(r_{ij})_{1 \leq i,j\leq n}$ be a family of independent identically distributed random variables which take the values $-1$ and $1$ with ...
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116 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
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99 views

How many independent random variables can be defined over a standard probability space.

Consider the probability space $(0,1)$ equipped with the Borel $\sigma$-algebra and the uniform-distribution as the probability. Is there a set of independent random variables defined on it with the ...
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49 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
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89 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 ...
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109 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} ...
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159 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 ...
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127 views

Measurability of number of upcrossing $U_I(\alpha,\beta; X)$ in continuous time

These definitions come from Karatzas and Shreve, Brownian Motion and Stochastic Calculus. We may take for granted that $U_F(\alpha,\beta; X(\omega))$, the number of upcrossings over $[\alpha,\beta]$ ...
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29 views

How to simulate a sequence of partial sums $(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),$ given some properties.

I want to generate/simulate a sequence of partial sums. $$(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),\text{ for }1 \leq n \leq 100$$ Let $W$ be a random variable such that: $W \thicksim ...
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66 views

Is there a conditional version of the asymptotic equipartition property?

Let $X_i$ be independent random variables with $\operatorname{Pr}(X_i = x) = p_x$, and let $F_n$ be the empirical frequency distribution of $X_1, \ldots, X_n$: that is, $(F_n)_x$ For any frequency ...
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215 views

History of odds making in sports betting

Can anyone provide a reference to the history of odds making in sports betting? In many cases, certain odds are set and then adjusted as people make bets. However, I am having difficulty tracing the ...
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97 views

A question about the stability of a property of the normal distribution

Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
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107 views

a simpler proof for an inequality in probability

I am trying to show that for any two real $X,Y$ iid there holds that $$ P(|X - Y| \le 2) \leq 3P(|X - Y| \le 1) $$ I am getting nowhere with this and so I did some digging and found the following ...
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173 views

Relation between factor graph and conditional probability distribution

First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me. The question Let say I have a factor graph illustrated in the figure. The ...
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64 views

Convex Hull of Sampled Random Points

Consider $N$ i.i.d. random points $x_1,x_2,..., x_N\in \mathbb{R}^n$, sampled from a given distribution $d$ defined on $\mathbb{R}^n$. Let $\mathcal{C}_x \subset \mathbb{R}^n$ be the convex hull of ...
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50 views

Do we have homogeneous coordinates for probabilities?

As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, ...
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84 views

measurability and convergence of unconditional probability measure

This should be simple, and with a bit of study I should get there, but I am very tired and in a need for a hand. Thanks in advance. If $\rho$ is a probability measure over $Y$. For each $y \in Y$ ...
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71 views

Is there a canonical probability measure on smooth curves?

For continuous curves, we have Brownian motion giving the most natural probability measure. However, the sample paths of Brownian motion are almost surely terribly behaved (not of bounded variation, ...
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231 views

Convergence of Martingale.

The question is: 5.2.11. Let $X_n$ and $Y_n$ be positive integrable and adapted to Fn. Suppose $\mathbb E(X_{n+1}|\mathcal F_n) ≤ (1 + Y_n )X_n$ with $ \sum Y_n < \infty$ a.s. Prove that ...
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114 views

Narrow convergence determining classes of sets

Notation used: $S$- Seperable metric space $\mathcal{C}_u(S)$- Class of uniformly continuous functions from $S$ into $\mathbb{R}$ $\mathcal{P}(S)$- Space of probability measures on ...
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106 views

maximum renewal rate of a Markov chain

Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ ...
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126 views

Relation between maximizer's derivative and maximizing function

Let $u(x)$ be a continous bounded function such that $u'(x) >0$, $u''(x) < 0$. Define $A(x) = -\frac{u''(x)}{u'(x)}>0$. Let $Y$ be a random variable with $\mathbb{E}|Y|<\infty$. I solve a ...
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124 views

Failure criteria for a collection of independent evolving discrete random variables

I have built a computer model that contains a collection of independent discrete random variables. They each have values of between $0$ and $k$ where $k$ is between $4$ and $15$ and is constant for ...
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46 views

higher order uncertainty

Context: As a general rule of thumb parameter estimation (in data analysis) works better the fewer the biased assumptions one makes. For instance, when presented with a coin and trying to estimate the ...
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105 views

about birth and death process

My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
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371 views

The application of Doob's inequality and Doob's decomposition theorem

1) What is the application of Doob's inequality? Can we use Doob's inequality ($L^1$) to prove the convergence (maybe almost surely) of a martingale? Doob's inequality: Let $X$ be a submartingale ...
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300 views

Does Kolmogorov 0-1 law apply to every translation invariant event?

Let $\Lambda$ be a lattice in $\mathbb{R}^d$, $d \ge 2$, thought of as an infinite graph. In percolation theory we consider properties random subgraphs of $\Lambda$. In site percolation $\Lambda^s_p$ ...
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656 views

When can we interchange the derivative with an expectation?

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Is it true in general that $ \frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t)) $? If not, ...
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806 views

Uniform distribution on the unit circle (in the complex plane)

I was trying to prove that for a standard complex Gaussian variable $Z$ it holds that $|Z|^2$ is exponentially distributed with parameter 1, $\frac{Z}{|Z|}$ is uniformly distributed on the unit circle ...
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66 views

Suboptimal confidence for bounds on the extreme eigenvalues of random matrices: can they be improved?

Typical results from the literature of Random Matrix Theory (RMT) (e.g., Tropp'11, Vershynin'12) provide probabilistic guarantees for large deviation of the extreme eigenvalues of random matrices like ...
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482 views

Convergence in Probability to infinity

i was lately reading the book of Kallenberg "foundations of modern probability". I have a problem with understanding one of this thoughts(p. 70, Theorem 4.17): Let $\xi_1,\xi_2,\ldots$ be independent ...
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276 views

Is every killed Markov process still a Markov process?

Suppose we've got $X=(X(t))_{t\geq 0}$. $X$ is a strong Markov process with respect to filtration $\mathcal{F}_t$, taking values in some subset of $\mathbb{R}$. We take $\tau$ - a stopping time w.r.t ...
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276 views

Cover time and intersection time for lazy random walks on graphs

Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p_{ij}=1/(2d(i))$ where $d(i)$ is the ...
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173 views

Easy probability theory proof

$A$ and $B$ are random occurrences in $\Omega$. Prove that if $P(A)=0{,}9$ and $P(B)=0{,}7$, then $P(A\cap B')\leq0{,}3$, where $B'$ is a complementary event of $B$. I thought of something like this: ...
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149 views

Singular measures - approximate characteristic function

One can decompose a $\sigma$-finite measure $\mu$ on $\mathbb{R}$ in three parts: $\mu_{ac}$: absolutely continuous $\mu_{sc}$: singular continuous $\mu_{pp}$: pure point A common example for a ...
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173 views

Bayesian inference on partitioned multivariate Gaussian

My question is on Bayesian inference of partitioned multivariate Gaussian. To make things easier, suppose there is a 2-dimensional Guassian, $$ X_1 \sim N(\mu_1, \sigma^2_1) \\ X_2 \sim N(\mu_2, ...
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677 views

Definition of convergence in distribution

My question is convergence in distribution seems to be defined differently in Wikipedia and in Kai Lai Chung's book. My view is that the one by Wikipedia is a standard definition of convergence in ...
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152 views

How to solve equation involving binomial coefficient?

I'm reading this paper which says If we have $$ \binom n d p^{\binom d 2} = 1 $$ where $ 0 < p \le 1$, then $$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) ...
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152 views

Convergence of marginal distribution in the case of convergence in distribution

Suppose a random variable $Z_n=(X_n,Y_n)$ takes a value on $A= [0,1] \times B=[0,1]$. We can consider a conditional distribution of $Y_n$ given the realized value of $X_n$. Now suppose $Z_n$ converges ...
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278 views

An unconscious statistician's higher central moments

Recall the 'law of the unconscious statistician' (wikipedia), namely that one can calculate the expectation of the transform $g(X)$ of a random variable $X$, given the transformation and the ...
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159 views

An identity about a continuous local martingale

Let $M_t$ be a continuous local martingale with $M_0=0$ and define $I_t^0=1$ and $I_t^n= \int_0^t I_s^{n-1}\; dM_s$. Prove that we have $$n I_t^n= I_t^{n-1}- \int_0^t I_s^{n-2} \;d((M)_t) $$ where ...