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

Respond to the three interview questions below.

1) Basic properties of a Markov (transition) matrix

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

• List the properties that define $P$ (stochasticity, non-negativity, etc.).
• List the key linear-algebra facts that always hold (e.g., eigenvalue facts, conditions for existence of a stationary distribution).
• Give a short account of the conditions under which $P^k$ converges as $k\to\infty$ and what it converges to.

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

Suppose 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 including an intercept. Next 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, so $R^\top R=I$). You refit OLS using $x_i^*$ (with an intercept).
• What is the relationship between the estimated coefficients $\hat\beta$ and $\hat\beta^*$?
• Do the fitted values $\hat y$ remain unchanged under 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

Players A and B each flip a fair coin every round, at the same time.

• A’s stopping time $T_A$ is the first round $t\ge 2$ in which A has obtained two consecutive heads (that is, A has $H$ in rounds $t-1$ and $t$).
• B’s stopping time $T_B$ is the first round $t\ge 1$ in which B flips a tail. Compute $\mathbb{P}(T_A < T_B)$. (If both events occur in the same round, that does not count as “A before B”.)

Overview: This multi-part question tests understanding of Markov chain and stochastic matrix properties, linear-algebra facts about eigenvalues and stationary distributions, invariance and sampling-distribution properties of OLS under orthonormal rotations, and discrete stopping-time probability reasoning for coin-flip processes.