Solve Markov, OLS-rotation, and coin-toss probability

Answer the following three interview questions.

1) Basic properties of a Markov (transition) matrix

Let $P$ be a transition matrix of a (time-homogeneous) Markov chain on $n$ states.

• State the defining properties of $P$ (stochasticity, non-negativity, etc.).
• State key linear-algebra properties that always hold (e.g., eigenvalue facts, stationary distribution existence conditions).
• Briefly describe conditions under which $P^k$ converges as $k\to\infty$ and what it converges to.

2) OLS properties after a rotation (two independent uniforms)

You observe i.i.d. data $(x_{i1}, x_{i2}, y_i)$ for $i=1,\dots,n$, where

• $x_{i1}, x_{i2}$ are independent and identically distributed $\mathrm{Unif}(-1,1)$ (mean 0),
• the data follow a linear model $y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \varepsilon_i$ with $\mathbb{E}[\varepsilon_i\mid x_{i1},x_{i2}]=0$ and $\mathrm{Var}(\varepsilon_i\mid x_{i1},x_{i2})=\sigma^2$ .

You fit OLS with an intercept. Now define a rotated feature vector

$x_i^* = R\,x_i,\qquad x_i = (x_{i1},x_{i2})^\top,$

where $R$ is a 2D rotation matrix (orthonormal: $R^\top R=I$). You refit OLS using $x_i^*$ (with an intercept).

• How do the estimated coefficients transform (relationship between $\hat\beta$ and $\hat\beta^*$ )?
• Are the fitted values $\hat y$ invariant to this rotation?
• Under what conditions (on the design distribution) is the sampling distribution of the slope estimator “rotation-invariant” (at least in second moments)?

3) Two players flipping coins in parallel

A and B flip fair coins simultaneously each round.

• A’s stopping time $T_A$ : the first round $t\ge 2$ at which A has flipped two consecutive heads (i.e., A has $H$ at rounds $t-1$ and $t$ ).
• B’s stopping time $T_B$ : the first round $t\ge 1$ at which B flips a tail .

Compute $\mathbb{P}(T_A < T_B)$. (If both events happen in the same round, that is not counted as “A before B”.)