# Tagged Questions

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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### What does it mean to integrate a Brownian motion with respect to time?

I am reading about stochastic process, but could not make sense if one equation I encountered. Can anyone help me understand it? The equation states that suppose R(s) is an interest rate process, ...
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### Probability of an Ornstein-Uhlenbeck process

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, with $\tau < \infty$. The following definition is ...
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### How to rotate an $n$-dimensional normal distribution, to maximize the likelihood of a sample

Suppose we have a normal distribution with a diagonal covariance matrix S and mean $0$, i.e. $N(0,S)$. I want to find a Rotation matrix $R$, to rotate this distribution to maximize the likelihood of a ...
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### How to provide Mathematical Proof for number theory scheme?

I have a set S={1,2,...,N-1}. N=pq (where p and q are RSA prime numbers). Scenario is that User need to retrieve the Database blocks without revealing his block index to the Server i.e, Private ...
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### How often was the most frequent coupon chosen?

In the coupon collector's problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how ...
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### (Infinite hat)-guessing problem

$2$ men are playing a game: they are wearing countably infinitely many hats on their heads. The hats are either black or white with probability $\frac 12$. They see the other's man hats but cannot see ...
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### Lipschitz constant/derivative of the stationary distribution of a Markov chain under perturbations in the transition kernel

I'm interested in the following question: Given a parameter $t\in \mathbb{R}$ and a column stochastic matrix $P(t)$ (i.e., $e^T P(t)=e^T$ and $P(t)_{ij}\ge 0$), calculate the Lipschitz constant of ...
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### Is statistical physics background desirable for probability theory?

I am talking about higher probability viz. Brownian Motion, Ergodic Theory, Concentration, Percolation, Random Graphs, Random Matrix, etc. Going through books, I find that somehow or the other, many ...
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### What is the “time change” of an adapted finite-variation stochastic process?

Let $(\Omega, \mathcal F,\mathbb P)$ be a probability space equipped with a filtration $\{\mathcal F_t:t\in\mathbb R_+\}$ satisfying the usual conditions of completeness and right-continuity. Suppose ...
Assume you have a probability space $(\Omega,\mathcal{F},P)$, and you have a filtration $\{\mathcal{F}_t\}$ and a stopping time $\tau$. Then all the books I have seen define the stopping time sigma-...