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 ...

learn more… | top users | synonyms (1)

3
votes
0answers
93 views

Why are independent random variables always uncorrelated?

I'm trying to show that, if $X$ and $Y$ are two independent random variables, then $\mathbb E(XY)=\mathbb E(X)\mathbb E(Y)$. For the purposes of this question, assume that $X$ and $Y$ are ...
3
votes
0answers
124 views

interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal? $$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} ...
3
votes
0answers
42 views

Normal random variables, joint distribution

If I have two normally distributed random variables, is their joint distribution always elliptical, i.e. fully characterized by a mean vector and variance-covariance matrix?
3
votes
0answers
53 views

Show equivalences concerning independence

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. We say, that $(E_i\in\mathcal{A}:i\in I)$ is a family of independent events, if for any finite subset $I_0\subset I$ it is ...
3
votes
0answers
34 views

Number of times above a linear boundary for a finite variance random walk

I consider a random walk $(S_n)$ with mean zero and finite variance, and $\epsilon>0$. Is it true that $$ \mathbb{E}\left[\sum_{n=0}^{+\infty} 1_{S_n>n\epsilon}\right] < +\infty \quad ? $$ ...
3
votes
0answers
69 views

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers ...
3
votes
0answers
62 views

Convolution-like operator on (probability) measures on $[0,1]$ yielding measures on $[0,1]$.

Is there a "correct" or "best" way to define convolution of two (Borel) probability measures on $[0,1]$ to yield another probability measure on $[0,1]$? Recall that the convolution, $\mu * \nu$, of ...
3
votes
0answers
161 views

Independence in infinite sequence of random variables

I was reading this page from planetmath. It states the following, Consider an infinite sequence of independent random variables $\{X_n,n\in \mathbb{N}\}$. Using the Monotone Class Theorem one ...
3
votes
0answers
165 views

Explanation of Radon-Nikodym derivates wrt to probabilities

I am currently working in communications, where a lot of work is done via probability calculations (densities and such). As I am not a mathematician, I do have a quite hard time understanding one ...
3
votes
0answers
62 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
3
votes
0answers
230 views

Gambler's Ruin Problem

Two gamblers, A and B, initially have capital 7 and 13 respectively. At each round of the game, player A wins 1 from B with probability p, or loses 1 to B with probability q = 1 - p. Assume that 0 ...
3
votes
0answers
52 views

Connection of weak and pointwise convergence

Suppose $Y_n:=v_n^{-1}\sum_{i=1}^n X_i\xrightarrow{}Z\sim N(0,1)$ in distribution where $(X_n)_{n\in\mathbb{N}}$ is a stationary sequence of real random variables with finite variance and $v_l$ is a ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
71 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 $n\to\infty$. ...
3
votes
0answers
116 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 ...
3
votes
0answers
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 ...
3
votes
0answers
143 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 ...
3
votes
0answers
109 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: ...
3
votes
0answers
265 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 ...
3
votes
0answers
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. ...
3
votes
0answers
81 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 ...
3
votes
0answers
119 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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} ...
3
votes
0answers
162 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 ...
3
votes
0answers
129 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 $[\alpha,\beta]$ ...
3
votes
0answers
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 ...
3
votes
0answers
67 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 ...
3
votes
0answers
218 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 ...
3
votes
0answers
98 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 $X$, $Y$ ...
3
votes
0answers
108 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 ...
3
votes
0answers
178 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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, ...
3
votes
0answers
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 $y \in Y$ ...
3
votes
0answers
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, ...
3
votes
0answers
237 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 ...
3
votes
0answers
115 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 ...
3
votes
0answers
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 $\mathfrak{S}$ ...
3
votes
0answers
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 ...
3
votes
0answers
126 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
372 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 ...
3
votes
0answers
302 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 $\Lambda^s_p$ ...
3
votes
0answers
679 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, ...
3
votes
0answers
812 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 ...