# 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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### Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

This is a verification request for a counterexample that I think I have found for Problem 32.13 on page 427 in Patrick Billingsley’s Probability and Measure textbook (third edition, but the problem ...
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### 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 ...
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### Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random Variable)...
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### Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
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### Is there a well-defined `uniform' distribution on $C([0, 1])$?

I'm wondering whether we can define a uniform distribution on the space of continuous functions over a compact set, e.g. $C([0, 1])$. If so, then how should I rigorously describe it? And how can I ...
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### Conditions for the existence of a density with respect to Lebesgue measure

Let $X:\Omega \to \mathbb{R}$ be a random variable on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and denote by $$\chi(\xi) := \mathbb{E}e^{i \xi \cdot X}, \xi \in \mathbb{R}^d,$$ its ...
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### Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
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### If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
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### Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
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### Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
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### Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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