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In the spirit of this answer to a different question, I'll offer a contrarian answer. How to understand probability theory from a structuralist perspective: Don't. To put it less provocatively, what …

The usual approach is to generate $n+1$ i.i.d. mean zero Gaussian random variables $X_1, \dotsc, X_{n+1}$ to get a random point $X$ in $(n+1)$-space with rotationally invariant distribution and normal …

For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variab …

This old question and my old answer continue to get occasional attention, and I believe it's time for a new answer. Is there an introduction to probability theory from a structuralist/categorical per …

For your first question, the answer is yes, and I don't understand why it isn't better known since all the classical proofs of the central limit theorem generalize easily to that setting. See this sec …

There is a Hilbert space interpretation of independence, which follows from the interpretation of conditional expectation as an orthogonal projection, though it may be more complicated than you had in …

If by uniform measure you mean $(n-1)$-dimensional Hausdorff measure on the sphere, the answer is no. As a consequence of the results of this paper by Barthe, Csörnyei, and Naor, under mild regularit …

There is active interest in such results in high-dimensional geometry, and expander graphs have even been used explicitly as a tool. Take a look for example at this paper and the references on the se …

I'm guessing that you meant to write "real orthogonal matrices" without "symmetric", since the set of symmetric orthogonal matrices has Haar measure 0. Otherwise, please clarify what you mean by "a d …

You don't find much about time-inhomogeneous Markov chains because it's extremely difficult to prove anything about them without strong additional assumptions, and it's not clear what additional assum …

Without further assumptions you can't do better than the union bound (which should be $n e^{-\epsilon^2}$ as you've written things). If $X_i$ are identically distributed and the events $(|X_i| > \eps …

A little too long for a comment, but I don't have time right now to turn this observation into a proper answer: Another way to generate a uniform random point in the simplex is to let $b_1, \ldots, b …

You can do something with Talagrand's inequality for at least some normalizations. The simplest case would be if the $X_i$ are mean 0 and bounded, say $|X_i| \le 1$ almost surely, and you're looking …

The covariance is a multiple of the identity by simple symmetry considerations. For the constant, you just need, again by symmetry, and integration in spherical coordinates, $$ \mathbb{E} X_1^2 = \fr …

Any mixture of Gaussians has a density, which limits then sense in which a statement like you want to make can be true. The statement you propose doesn't make sense (in part) since a distribution is …