# Questions tagged [probability-theory]

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-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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### How to see that the Wick product has $0$ expectation.

In the book "Gaussian Hilbert Spaces" (Svante Janson) the author introduces the Wick product of a finite sequence of $n$ random variables living in a Gaussian Hilbert space $G$ as the orthonormal ...
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### Continuous time strong approximation (in the sense of Komlós--Major--Tusnády)

The Komlós--Major--Tusnády result asserts that, given a sequence of i.i.d. random variables $X_1,\dotsc,X_n$, one may extend the probability space and find normal i.i.d. variables such that the ...
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### Convolution of two Cardioid and vonMises PDFs

for the past few days, I've been "on and off" with Mardia's and Jupp's "Directional Statistics" to learn something new about approximating circular distributions. In particular, I've been looking at ...
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### Question about random variables convergence

Further going through old lecture notes I've stumbled upon this... Let's say we are dealing with a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $\sqrt{n}(X_n-1)\to N(0,2)$ in ...
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### Prove the existence of a finite additive continuation for measure

Prove that for any algebra $A$, every finitely additive measure defined on A has some finitely additive extension to the $σ$-algebra generated by $A$.
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### A line up question where the position and the number itself should not equal in an arrangement

There are $n$ numbers, from $1$ to $n$. Suppose that the numbers are arranged randomly, and asks there is no arrangement where the position and the number itself are equal. How many arrangements ...
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### Are “independent events” in probability really independent? [closed]

This is a hard and deep question. I understand very well the concept of independence. But, let us take two events: Event A (I throw a dice) and event B (some star explodes in an near galaxy). Are ...
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### Does joint distribution affect Radon-Nikodym derivative?

Given two real-valued random variables $X, Y$ with distributions $\mu_X, \mu_Y$. Suppose $\mu_X<\!<\mu_Y$, then the Radon-Nikodym derivative $\frac{d\mu_Y}{d\mu_X}(\cdot)$ exists $\mu_X$-a.e. on ...
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### Computation of the quadratic variation of Wiener Process.

My confusion arose from a commonly mentioned exercise: Show that the quadratic variation of Wiener Process is $\langle W\rangle_{T}=T$. Note that the quadratic variation here is the non-...
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### How do concepts such as limits work in probability theory, as opposed to calculus?

When I am flipping a fair coin and say that as the number of trials approaches $\infty$ the number of heads approaches $50\%$, what do I really mean? Intuitively, I would associate it with the ...
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### Relationship Between Entropy of a Measure and Hausdorff Dimension of Its Support

In this paper by Chhabra and Jensen, the authors make the claim (based on a theorem by Eggleston which is proven in this paper) that "for a special class of measures $\mu$ that arise from ...
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### Using Chapman-Kolmogorov Property to prove v=Qv

How would you use the Chapman-Kolmogorov property ($Q_{t+s}=Q_tQ_s$) to prove that v (a column vector distribution over the sample space) is a stationary distribution of Markov Chain $X_t$ with ...
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### Stationary Distribution of Ehrenfest Markov Chain

The example in my book for an Ehrenfest Markov chain is: A system of of two urns, A & B where there are 2n balls total in both urns. We are assuming that there are $i$ balls in urn A and $2n - i$ ...
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### Expectation of Inverse Normal CDF

Suppose a r.v. $\mu$ is distributed Normal $N(\theta,\sigma^2)$. Is there any way to derive the expectation $\mathbb{E}(\frac{\mu}{\Phi(\mu)})$ where $\Phi$ is the CDF of a standard Normal random ...
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### Poisson Arrival Process and Uniform Distribution

I'm brushing up on some basic probability and have this question: If we have a Poisson arrival process with arrivals $A_{1}, A_{2}, \dots$, and we know that there is one and only one arrival in a ...
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### Conditions for Weak and Strong law of Large Numbers

I am a second-year undergraduate student and was reading up the topics of the law of large numbers from Casella and Berger's Statistical Inference. They state the laws as follows: The link to the ...
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### Quadratic variation of true martingale

We know that for a continuous local martingale $M$ the quadratic variation $\left<M \right>$ is such that $M^2- \left< M \right>$ is a continuous local martingale. Is it true that if $M$ ...
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### $L^p$ Boundedness of a martingale

So I recently read a paper where the authors claim that if for some martingale $(M_t)_{t\geq 0}$ we have $$\mathbb E[M_{t+s}^p]-\mathbb E[M_s^p]\leq \exp(-cs) (\mathbb E[M_{t}^p ]-\mathbb E[M_t]^p)$$ ...
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### Stochastic processes - Why do we need filtration?

In the theory of stochastic process, besides the $\sigma$-algebra $\mathcal {F}$, we have an increasing sequence of $\sigma$-algebras $\{{\mathcal {F}}_{{t}}\}_{{t\geq 0}}$ called filtration. ...
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### If $\gamma$ is a coupling of $\delta_x$ and $\delta_y$, can we show that $\int f\:{\rm d}\gamma=f(x,y)$?

Let $(E,\mathcal E)$ be a measurable space, $\pi_i$ denote the projection of $E^2$ onto the $i$th coordinate, $\delta_x$ denote the Dirac measure on $(E,\mathcal E)$ at $x$ for $x\in E$ and $\gamma$ ...
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### Proving that $\sup E[X_n] \geq E[X]$

Consider the sequence of random variables $\{X_n\}_{n\geq1}$ such that $X_n$ are non-negative and $X_n \rightarrow X$ almost surely, with $\sup E[X_n]<\infty$. Prove $E[X]\leq \sup E[X_n].$ My ...
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### If $t\mapsto X_t$ is continuous almost everywhere and $(X_t)$ has independent increments, then $X_t - X_s$ follows a normal distribution?

The following statements can be found at Glasserman's Monte Carlo Methods in Financial Engineering. Given a stochastic process $(X_t)_{t\in [0,T]},$ if the mapping $t\mapsto X_t$ is continuous ...
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### Non-uniqueness in the $L^1$ martingale representation

Let $\xi \in L^1(P,\mathfrak F_T)$ on some probability space with measure $P$, supporting a Brownian motion, we consider the augmented filtration $\mathfrak F$ associated to $W$, and a time $T>0$. ...
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Consider the hypothesis: $X,Z$ random variables in $L_1$ and $Y$ bounded random variable. Can I prove the following result with the above hypotheses? If $E[XY]=E[ZY]$ (for each bounded r. v., Y) $\... 0answers 14 views ### Any normal random variable is ''essentially'' surjective Let$\Phi$be the cumulative distribution function of the standrad normal distribution. Denote$X: (0,1) \rightarrow \mathbb R$its inverse. Then$X$is a standard normally distributed random variable ... 0answers 17 views ### Negative moments of variables Let$f_i, i=1, \ldots, n$be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let$a \in R^n$. Find$E\left(\sum_{i=1}^nf_i a_i\...
say we sample a finite set a finite number of times (ie we have finitely many iid random variables $X_1,...,X_k$ taking values in a finite set). i read that the multiplicative chernoff bound can be ...
### A proof that $L^{\infty}$ is complete
I know this has been proved in other links, but I am wondering about the validity of the following proof: Suppose $X_n$ is a Cauchy sequence in $L^\infty$. Then there exists a subsequence \$Y_k \...