Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Proving $\sigma$-additivity and interchanging order of summation/integration just because positive

Prove $\sigma$-additivity in the ff: Let $\Omega = {\omega_1, \omega_2, ...}$ be some countable set. Let $\mathfrak{F} = 2^{\Omega}$. Consider a sequence {$p_n$} in [0,1] s.t. $\sum_{n=1}^{\infty} ...
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1answer
19 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
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1answer
18 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
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22 views

Can a biased physical random source be post-processed to control the bias?

Let $X_i$ with $i\in\mathbb N$ be a sequence of independent 6-ary random variables with distribution $\operatorname{Pr}(X_i=e)=p^e_i$ where $e\in\{1,2,3,4,5,6\}$ and $\sum_{e=1}^6p^e_i=1$. Let's ...
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1answer
36 views

I need help understanding this proof about convergence in distribution

The proof says that we used the fact that $(1-\epsilon)^\frac{x}{\epsilon} \rightarrow e^{-x}$ Why is this so? How do I prove this? Also, why do we need the fact that $\lfloor x/p_n \rfloor - ...
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1answer
12 views

Finding the unique Martingale Convergence Representation of a given r.v.

According to the martingale representation there exists a unique $g(t,\omega) \in \mathcal{V}(0,T)$ such that $M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T]$ Find g in the case ...
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0answers
39 views

Median-median inequality

An elementary result from Chebyshev's theorem is that the median and mean of a random variable do not differ by more than one standard deviation. I'm curious if there is a similar result for ...
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17 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
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1answer
17 views

Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...
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2answers
22 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
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0answers
16 views

Upper Bound on Mutual Information

I am interested in an upper bound on mutual information that I have been encountering frequently in the statistics and probability literature. I have yet to see the "purest" form of the inequality, so ...
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0answers
10 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
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1answer
57 views

Convergence in Probability (weak law of large numbers) [on hold]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d. standard normal random variables. Prove that $$\frac{X_1X_2 + X_2X_3 + X_3X_4 + \cdots + X_{n−1}X_n}{n}$$ converges in probability to 0. I started this by ...
2
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1answer
97 views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in\{-\alpha,\alpha\}$, $Y\in\{-\alpha,\alpha\}$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
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7answers
28k views

What is the best book to learn probability?

Question is quite straight... I'm not very good in this subject but need to understand at a good level.
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25 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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1answer
26 views

martingale difference

I am trying to solve the following question. {$ξ_k$} is $F_n$-martingale difference (i.e. for every $n$, $E[ξ_n|F_{n-1}]=0 $ a.s. ) Also, for every $n$ , $E[ξ_n^2]<\infty$ Show that ...
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1answer
16 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim Bin(n_i, p_i)$. We know that $$Pr[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is there an easy way to ...
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1answer
81 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
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0answers
48 views

Qual Question concerning martingale

Suppose $X_n$ is a sequence of random variables that has the property that $\sup|X_n| \leq 1$ a.s. Then use Doob's decomposition to prove that $\sum_{n\geq 1} X_n$ converges a.s. iff the sum ...
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0answers
13 views

When do almost all random variables attain the expectation?

Given some sample space $\Omega$ I choose uniformly at random some $X \in \Omega$. Assume that I know the expected value $\mathbb{E}(X)$. What are the further conditions I need if I want to talk ...
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0answers
5 views

measure of dependence for copula

I have some question about the paper of Schweizer and Wolff (1981). The question concerns about the following bound $$\int_0^1\int_0^1|C(u,v)-uv|\,du\,dv\leq\frac{1}{12}$$ where $C$ is any copula. ...
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0answers
31 views

Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
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1answer
15 views

Conditional variance and expectation of random variables

I am trying to show whether the following statement is true or not: $E(X^2|A)E(1|A) \ge E(X|A)^2$ It is straightforward to prove this statement if $X$ was not conditioned on event $A$. Because, one ...
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1answer
217 views

What is the probability of picking Exactly 1 red marble and than not 1 red marble? without rep.

A urn has 3 red marbles, 2 blue marbles, 1white, 1 black 1 brown. What is the probability of getting exactly 1 red marble than not 1 red marble? What is the probability of getting at least 1 red ...
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0answers
14 views
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0answers
11 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
2
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1answer
114 views

Is Wishart Matrix?

Analyzing a system, I have faced a problem which is related to Random Matrices and in particular Wishart matrix. The problem is as follows: Lets assume $\boldsymbol{H}$ is an $m\times n$ random ...
3
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1answer
53 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
2
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0answers
15 views

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian. Without loss of generality we may assume that $X$ and $Y$ are ...
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27 views

Qual martingale theory question

Suppose $Y,X_{1},...,X_{n},\ldots$ have the following properties: $Y$ has the exponential distribution. That is, $P(Y>t)=\exp(-t)$ for $t>0$; Conditionally on $Y, X_{1},\ldots,X_{n}$ are i.i.d. ...
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3answers
45 views

Independence of $X$ and $2X$

Are these two random variables independent? Unfortunately, I don't know probability theory enough to answer this question. I know for a fact that if $X$ and $Y$ are independent random variables and ...
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0answers
22 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
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35 views

Are A and B always dependent? [on hold]

Let $A$ and $B$ be proper subsets of $\Omega$, with $A\cup B=\Omega$. Are $A$ and $B$ always stochastic dependent and why?
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21 views

Number of trials for two successes

Let Y denote the number of successes required for two successes in a series of Bernoulli trials with parameters p and q. I want to know whether $P[Y=n]=\binom{n}{2}p^2q^{n-2}$ or ...
6
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1answer
223 views

What is the most extreme set 4 or 5 nontransitive n-sided dice?

A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia) For some sets, the ...
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2answers
42 views

recurrent events-Probability of even number of successes

Let E be the event of an even number of successes. $u_n$:Probability of E occurring at the nth trial not necessarily for the first time $f_n$:Probability of E occurring at the nth trial for the first ...
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0answers
11 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
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1answer
49 views

Probability to pick an integer number $k\in\mathbb{Z}$ from the field $\mathbb{R}$

In the field of the real numbers $\mathbb{R}$, we build a subset $A\subset\mathbb{R}$ $A=\{k\}, k\in\mathbb{Z}, k=-N,-N+1,...,0,1,2,..,N$. If we pick infinite times a number $x$ from $\mathbb{R}$ with ...
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0answers
10 views

How to Prove this Multinomial Distribution Inequality

I have the following lemma, but there seems to be one (or two) mistakes in the proof found in this paper (lemma 3). The lemma states that for $Multinomial(n,p_1,\ldots,p_k)$ distributed ...
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1answer
53 views

Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
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0answers
8 views

About the space $((R^n)^{[0,\infty)},\mathcal{B},\tilde{Q}^x)$ in Oksendal SDE book

I am reading the book Stochastic differential equations (6th ed.) by Oksendal. I am not sure about the meaning on P.146. (Below Theorem 8.3.1) It says that ``if we identify each $\omega \in \Omega$ ...
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2answers
23 views

Reliability Probability problem

What is the Probability that at least one close path is formed from A to B where each switch has a Probability of close = p and each switch acts independent of the other Proposed Solution Let ...
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Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
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11 views

Cesaro limit of stochastic matrices [on hold]

For a Markov chain, we can write the transition matrix as $$ P = \left( \begin{matrix} Q & R_1 & R_2 & \cdots & R_h \\ 0 & B_1 & \mathbf{0} & \cdots & \mathbf{0} \\ 0 ...
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1answer
14 views

Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
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1answer
32 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
2
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1answer
144 views

Snell envelope and optimal stopping time

Suppose $(G_n)_{0\leq n\leq N}$ is a process adapted to a filteing $(\mathcal{F}_n)_{0\leq n\leq N}$. The Snell envelope of $(G_n)$ is the smallest supermatingale dominates $(G_n)$. It's defined as ...
2
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0answers
31 views

Sequence of independent RV

Let $(X_n)$ be a sequence of independent RVs that converges in probability to some RV $X$. Show that $X$ is constant a.e. My attempt So from definition of convergence I have: $$\forall\epsilon>0 ...
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1answer
18 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...