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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126 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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65 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
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214 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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95 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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166 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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224 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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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 ...
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123 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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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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365 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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294 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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645 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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799 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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476 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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274 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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270 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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172 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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147 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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662 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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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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365 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 ...
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455 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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86 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 ...
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20 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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24 views

Is there a connection between uniform law of large number and Ibragimov's conjecture?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, ...
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21 views

Distribution of $f(x,|h|)$, being $|h|$ rayleigh distributed

INTRODUCTION Let's supose we receive the following signal: \begin{equation} y[n] = hx[n]+W[n] \end{equation} where: $x[n] = Ae^{j2 \pi f_c t}$ is the transmitted signal $f_c$ is the carrier ...
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39 views

The polynomial is dense in $L^2$ with non-lebesgue measure

Assume the function $u\to \mathbb E[e^{iuX}]$ is analytic in a nbhd of $0$ where $X$: $\Omega\to \mathbb R$ is a random variable. Now I want to conclude that the space of polynomial, denoted by ...
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32 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
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15 views

What does the s-Transform (exponential transform) mean conceptually? What does it show us?

I don't understand the conceptual idea. If I have PDF, and I calculate its s-transform for some s, what do I know that I did not know before?
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10 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
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79 views

Moment generating function of $(W_T, \max W_t)$

Does there exist an explicit formula for the moment generating function $\psi(u, v) = E e^{u W_T + v M_T}$ of the pair $(W_T, M_T)$ where $M_T = \max_{0\leq t\leq T} W_t$? Using the well-known pdf of ...
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71 views
+50

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
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34 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
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42 views

Electrostatic capacity of two spheres with changing radii

Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits. My question is the following (in a simplified setting). All this ...
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29 views

Brownian motion, find minimum of function

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \geq 0)$ a Brownian motion and $(\mathcal{F}(t),t \geq 0)$ its natural filtration. Suppose $0 \leq s \leq t$ and let $f:\mathbb{R} ...