1 (U1)

R fills matrices columnwise by default. Specifying rowwise filling gives the requested order:

A <- matrix(c(1/3, 2/3, 0,
              1/4, 1/2, 1/4,
              0,   1/4, 3/4), nrow = 3, byrow = TRUE)

The result is a transition matrix, with nonnegative entries and every row summing to one.

Use inverse transform sampling for the categorical draw. Validate a finite nonnegative probability vector, allowing only a small rounding tolerance in its total, then compare a single uniform draw with the cumulative probabilities:

sim1 <- function(p) {
  if (!is.numeric(p) || length(p) == 0L ||
      any(!is.finite(p)) || any(p < 0) ||
      abs(sum(p) - 1) > 1e-12) {
    stop("p must be a nonempty finite probability vector")
  }
  p <- p / sum(p)
  cumulative <- cumsum(p)
  cumulative[length(p)] <- 1
  u <- runif(1)
  as.integer(which(u < cumulative)[1L])
}

The returned value is

j

when

u

falls in the interval between the previous cumulative sum and the

j

th one. That interval has length

p_{j}

, so the output has the required probabilities. Strict comparison skips zero-probability categories even if a simulated uniform value equals an interval boundary. Exactly one call to runif(1) is made.

c

 0  0

Solution

 0  0

The function below counts the initial uniformly chosen state as the first of the

n

visited states, and then generates

n - 1

transitions using the appropriate row of

P

simchain <- function(n, P) {
  if (!is.numeric(n) || length(n) != 1L || !is.finite(n) ||
      n < 1 || n != floor(n)) stop("n must be a positive integer")
  if (!is.matrix(P) || !is.numeric(P) || nrow(P) == 0L ||
      nrow(P) != ncol(P) || any(!is.finite(P)) ||
      any(P < 0) || any(abs(rowSums(P) - 1) > 1e-12)) {
    stop("P must be a finite square stochastic matrix")
  }
  k <- nrow(P)
  state <- sim1(rep(1/k, k))
  counts <- numeric(k)
  counts[state] <- 1
  if (n > 1) {
    for (step in seq_len(n - 1)) {
      state <- sim1(P[state, ])
      counts[state] <- counts[state] + 1
    }
  }
  counts / n
}

Conditional on the current state