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 [tag:probability-...

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10 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
3
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14 views

Factory inspections on a budget

A factory inspector is testing the efficiency of $n$ machines. To pass the inspection, each machine is required to run at or above a certain standard efficiency. The inspector can measure the ...
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0answers
18 views

ergodic theorem for expectation of positive recurrent diffusion

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{...
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1answer
39 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
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0answers
27 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
3
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1answer
34 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
2
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0answers
32 views

One question regarding independence of $\pi$ systems.

Let G and H be sub-sigma-algebras of F and I and J are $\pi$ systems such that $\sigma(I)=G$ and $\sigma(J)=H$ Can anyone explain the following quote? Suppose I and J are independent, for fixed ...
4
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5answers
528 views

Intuition behind Chebyshev's inequality

Is there any intuition behind Chebyshev's inequality or is that only pure mathematics? What strikes me is that any random variable (whatever distribution it has) applies to that. $$ \Pr(|X-\mu|\...
1
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1answer
18 views

Voting with 3-way ties

From Peter Winkler's 'Mathematical puzzles' Ashford,Baxter and Campbell run for election and end up in a 3-way tie. To break it, they solicit voters' second preference and there is also a 3-way tie. ...
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2answers
37 views

X random variable in $\mathbb{N}$ independence of events

If I have a random variable $X$ with values in $\mathbb{N}$, $$\mathbb{P}(X=n)=\frac{1}{n^s\zeta(s)}$$ where $s>1$ and $\zeta$ the Riemann zeta function, then how can I show that $$A_i=E_{p_i^2}=\...
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0answers
39 views

What is the almost sure limit of the normalized sum of these random variables? [on hold]

What is the almost sure limit of $\displaystyle\sum_{i=1}^n \frac {X_i} n$ if $$\displaystyle\mathbb{P}(X_n=n^2)=\frac 1 {n^2}$$ and $$\displaystyle\mathbb{P}(X_n=0)=1-\frac 1 {n^2}?$$ My guess is ...
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0answers
20 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
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0answers
11 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
1
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1answer
34 views

$X_1, X_2, \dots$ uncorrelated, $\frac{Var[X_i]}{i} \rightarrow 0$, then $\frac{S_n}{n} - \frac{\mathbb{E}[S_n]}{n} \rightarrow 0$ in $L^2$

Let $X_1, X_2, \dots$ be uncorrelated random variables with $\mathbb{E}[X_i]= \mu_i$ and $\displaystyle\frac{Var[X_i]}{i} \rightarrow 0$, when $i \rightarrow +\infty$. Show that $\displaystyle\frac{...
1
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1answer
52 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
1
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1answer
9 views

Independent events from any other

In $(\Omega,\mathcal{F},P$) probability space, how can I show that $\forall A\in \mathcal{N}=\left\{ A\in\mathcal{F}: \mathbb{P}(A)=0,or\,\, \mathbb{P}(A)=1 \right\}\Rightarrow$ $\forall E\in\Omega$...
2
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1answer
37 views

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$?

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$? For context, I'm re-reading Kallenberg and in Chapter 3, on page 49, in his proof of Lemma ...
0
votes
1answer
26 views

Sum of Random Variables i.i.d. with $\mathbb{E}[|X_n|]=+\infty$

Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $\mathbb{E}[|X_n|]=+\infty,\forall n$. Prove that $\sum_n \mathbb{P}[|X_n|>kn]=\infty$ with $k\...
2
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1answer
34 views

Transformation of random variables that preserves the distribution

Suppose we have a random variable $X$ with distribution $F_X$. Let $X_1$ and $X_2$ be two independent copies of $X$. My question: can we find a transformation $Z=g(X_1,X_2)$ such that the ...
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0answers
32 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
-1
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0answers
25 views

conditional expected value and not mutual indipendent events [on hold]

$\newcommand{\P}{\mathbb{P}}$ Let be $E,G,H$ pairwise independent events but not mutual (e.g. $P(E\cap H)=\P(E)\mathbb{P}(H),\,\P(G\cap H)=\P(G)\P(H), ...but \,\P(E\cap G\cap H)\ne\P(E\cap G)\P(H)$ ...
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2answers
99 views

Probability Of four events

I have an issue with calculating probability of union of four events, formula listed below \begin{align*} P(A \cup B \cup C \cup D) & = P(A) + P(B) + P(C) + P(D) - P(A \cap B) - P(A \cap C)\\ &...
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2answers
183 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
1
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1answer
44 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
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0answers
54 views
+50

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
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0answers
18 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Let $X_t^x$ be a solution to the SDE: $dX_t=b(X_t)+\sigma(X_t)dW_t$ with $X_0=x$. Assume that $b$ and $\sigma$ are Lipschitz Continuous. I want to proof that there exists $0<L$ such that $E\left|...
0
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1answer
48 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
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1answer
60 views

Probability for a random permutation

Given a set of $N$ elements and a uniformly distributed random number generator, which always generates values between $0$ and $N-1$. Then the probability to get a random permutation (without re-draws)...
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0answers
18 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
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0answers
20 views

Expectation of Martingale [on hold]

I am reading an article of American Put option here and I want to understand the proof of Theorem 1 (Main Decomposition of the American Put). It is stated that: $$\exp(-rT) \max[0,K-S_T]=P_0 - rK\...
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0answers
17 views

probability measures vs. probability distributions vs. measure of probability density

I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct: Suppose we have a ...
1
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1answer
21 views

Weak convergence implies convergence on continuous functions

Let $X$ be a metric space, and let $\mu_n$ be a sequence of measures on $X$ converging weakly to a measure $\mu$, meaning for all bounded continuous functions $f$, we have $\int_{X}fd\mu_n \rightarrow ...
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2answers
53 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
1
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0answers
15 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
0
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2answers
17 views

Independence of Random Variables From Expectation Counter Example

I know that if $X$ and $Y$ are independent random variables, then $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$. I also know that the converse is not true, although I cannot seem to find an easy ...
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0answers
19 views

Communicating classes of a power of the irreducible transition matrix? [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P^k$. In terms of $d$ and $k$, how many communicating classes does $P^k$ have, and what is the period ...
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0answers
19 views

Convergence in law and sequence of reals

How to show that if I have $Y_n$ a sequence of random variables and $a_n$ a sequence of reals such that (i) $Y_n\to Y$ in law (ii) $a_n\to a$ i have $a_nY_n\to aY$ in law? I know that if I have (i)...
15
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2answers
9k views

Expectation of Minimum of $n$ i.i.d. uniform random variables.

$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)? I have conducted some simulations by Matlab, ...
14
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1answer
864 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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0answers
35 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
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1answer
75 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
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0answers
30 views

Entropy and Markov chain [on hold]

Assume that $X_n$ is a discrete Markov chain and $H$ is entropy function. I want to prove $$H\left(X_0\mid X_n\right) \geq H\left(X_0\mid X_{n-1}\right)$$ but I have no idea how to prove it. please ...
0
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2answers
69 views

Is it right to give equal chances?

In a certain town, the probability that it will rain in the afternoon is known to be $0.6$ Moreover, meteorological data indicates that if the temperature at noon is less than or equal to $ 25°C$, the ...
1
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1answer
23 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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0answers
24 views

Probability of $n$ numbers picked from set to have greater mean than set

If we have (N) varying non-negative numbers , with a mean equal to X, and a median less than X, if we pick (n) unique numbers from the set, what is the formula for the probability that the mean of ...
1
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0answers
63 views

Sum of random variables that are independent but not identical [closed]

For a real number $t$, let $X_t$ be the random variable that is uniformly distributed in the interval $[t/2, 3t/2]$. If $\{t_n\}$ is a sequence of positive real numbers, is there anything we can say ...
0
votes
1answer
170 views

Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent. [closed]

Let $Y_0, Y_1, ...$ be independent and identically distributed random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2$ for n = 0, 1, 2 ... Define random variables $X_n = Y_0Y_1Y_2...Y_n = \prod_{i=0}^...
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0answers
19 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
-3
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0answers
20 views