Vtyjkyuk

When bounding the convergence rate of many numerical linear algebra algorithms (e.g. the Lanczos method), typically one must use a bound on the coefficients of the initial vector in the eigenbasis of the input matrix. I wonder how to prove such a general bound for a uniformly randomly drawn vector from the unit sphere.

My idea so far is the following and I am wondering whether it makes sense:
Given a vector $v in mathbbR^n$ which is drawn random at uniform from the unit sphere, and some orthonormal basis $u_1,ldots,u_n$, I decompose this unit vector in this basis, i.e., $v = sum_i alpha_i u_i$. The question is now, whether I can lower bound the magnitude of these coefficients in probability, say with $p geq 1/2$, $alpha_i geq xi$.

My approach is the following very simple one:
Let $x$ be the vector with entries $alpha_i$;

1. It is straight forward to bound, based on Gaussian concentration bounds the magnitude of the norm of $v$ in high probability.


2. I could then conditionally on the norm being bounded find a lower bound on the entires, which are $chi^2$-distributed. However, I was not able to show a lower bound, since in particular I believe that the mean is $0$, which seems to imply that I can only obtain results of the form
   $mathbb P(min_i X_i > t) leq sum_i=1^n mathbb P(X_i > t) leq n cdot exp(-t^2/2 sigma^2)$ and then using that
   $min_1 leq i leq n |X_i| = min_1 leq i leq 2n X_i$ for $X_i+n = - X_i$ and hence obtaining:
   $$mathbb P(min_i |X_i| > t) leq 2n cdot exp(-t^2/2 sigma^2).$$
   However I would need in step three a different bound which lower bounds the probability.


3. If point (2) can be obtained, then I could just obtain the final success probability in terms of the error rates of the step (1,2) by bounding $mathbbP[v_i/||v|| geq c/sqrtn] geq 1 - delta_1 - delta_2$, where $delta_1/2$ are the results from the first two steps.

I was wondering if this approach is correct or if there is a better way?
Furthermore does anyone have a hint where I can find a lower bound on the minimum of a sub-gaussian distributed variable?