I have $k$ independent and identically distributed matrix valued random variables $X_{1}, ~X_{2}, \dots ~X_{k}$. The random matrices are all positive semidefinite with eigenvalues between $0$ and $1$.
I know that $\lambda_{1}(\mathbb{E}(X_{i})) \geq \alpha ~\forall~ i \in [k]$ where $\mathbb{E}(~.)$ is the expected value operator and $\lambda_{1}(~.~)$ extracts the largest eigenvalue of its argument. $\alpha$ is a positive constant (between $0$ and $1$).
I also know that $\lambda_{1}(\frac{1}{k}\sum_{i=1}^{k} X_{i}) \geq \alpha$ with high probability.
Can we say anything about $\lambda_{1} (X_{i})$ for any $i \in [k]$? More specifically, can we have a bound on the number of $i$'s for which $\lambda_{1}(X_{i}) \geq \alpha$?
Did you work more than $10$' about this question ?
How do you find a random matrix $A$ that is sym. with $spectrum(A)\subset [0,1]$ ?
For example, a sample for $A\in M_n$ is as follows:
i) Randomly choose (with uniform distribution on $[0,1]$) $n$ values $(a_i)$.
ii) Randomly choose a skew-sym. matrix $K$ and deduce (via the Cayley formula) an orthogonal matrix $U$.
iii) Put $A=Udiag(a_i)U^T$.
It seems to me that you are interested only by $E(\lambda_1(A))$, that is $E(max_i (a_i))$. The answer of this last problem is well-known.
A small effort my friend; come on !