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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3
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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 ...
3
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156 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
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0answers
60 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
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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
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0answers
83 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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70 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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215 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
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113 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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105 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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125 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
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122 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 ...
3
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0answers
78 views

Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
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359 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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284 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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623 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
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786 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 ...
3
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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 ...
3
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466 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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265 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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263 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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167 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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142 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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172 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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653 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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149 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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151 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 ...
3
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276 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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158 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 ...
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355 views

somewhere between sub-gaussian and central limit theorem

(problem statement revamped.) Definition: A random variable $X$ is a sub-gaussian if $$\Pr(|X|>t) \leq \exp(-t^2/k^2) \quad \text{for all $t\geq 0$ and for some constant $k$.}$$ I want to compute ...
3
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447 views

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, ...
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82 views

Multiplicative functionals for Markov process: discrete time

I read a theorem, stating Let $X_t$ be a Markov process w.r.t. to its natural filtration $(\mathcal F_t)$ on the space of cadlag functions on $\mathbb R_{\geq 0}$ and $(Z_t)_{t\geq 0}$ be a ...
2
votes
0answers
23 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
2
votes
0answers
19 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 ...
2
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0answers
15 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 ...
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0answers
9 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 ...
2
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0answers
39 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
36 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 ...
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0answers
22 views

Proving that a local martingale given by a stochastic integral is not a martingale

Let $X_t=\int_0^t e^{W_s^2}dW_s$ for $0\leq t\leq 1$ and show that is not a martingale. I guess the reason is that the expectation is not finite, but I'm not sure how to show it precisely. In fact ...
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21 views

Derivation of Gamma distribution characteristic function reference?

I was wondering if there was a derivation of the Gamma distribution characteristic function without expanding the $e^{itx}$ into an infinite summation?
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30 views

Problem on exchangeable events

$\newcommand{\p}{\mathbb{P}}$ $\newcommand{\tsum}{\textstyle{\sum}}$ $\newcommand{\N}{\mathbb{N}}$ Problem: Let $\left(A_{k}\right)_{k\geq1}$ be any infinite sequence of exchangeable event, i.e., ...
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0answers
34 views

Approximating the distribution of the infinte sum of random variables

What is the formal way to show that for an infinite sum of random varaibles $\sum_{i=1}^\infty X_i$, we have $\forall\varepsilon>0,\exists N<\infty$ such that $$0<P\left(\sum_{i=1}^N X_i\leq ...
2
votes
0answers
23 views

weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
2
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0answers
11 views

Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random ...
2
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0answers
42 views

Conditional density along a curve?

Let $X,Y$ be two random variables with $f_{X,Y}$ as their joint pdf and with $f_X(x)$ and $f_Y(y)$ as their marginal pdf's. The conditional pdf of, say $X$ with respect to $Y$ is $$f_{X\mid ...
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0answers
26 views

Complex moment problem

Let $X$ be a complex-valued random variable. For simplicity, let us assume that $X$ is bounded. If I know all of the $\mathbb{E}[X^k]$ then do I know the distribution of $X$? I know it suffices to ...
2
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0answers
9 views

Inward regular, not outward regular measure which is not locally finite

I want to find a measure, which is not locally finite, not outer regular but inner regular. Would the following example be correct? $\mu$ on the measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ ...
2
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0answers
27 views

Sufficient condition for a sequence of indepent events

Suppose $0 \leq p_n \leq 1$ and put $\alpha_n = \min(p_n, 1-p_n)$. Show that, if $\sum_n \alpha_n$ converges, then on some discrete probability space there exist independent events $A_n$ satisfying ...
2
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0answers
21 views

Conditioning a Brownian Bridge

Brownian Bridge: Consider the standard Brownian $X(t)$ motion conditioned to land at $b$ at time $1$. This means that for every trajectory {$(t,X(t)), 0 \leq t \leq 1$} of this process, its initial ...