Caminata Aleatoria

Es un proceso estocastico sobre los entros como espacio de estados.

\[ \psi_n = \sum_{i=1}^n X_i\]

Una variable aleatoria que depende de la posicion del proceso al cabo del tiempo \(n\).

\[ P(\psi_{n+1} = j | \psi_n = i) = \begin{cases} p & j = i + 1 \\ q & j = i - 1 \\ 0 & \text{en otro caso}\end{cases}\] Sea \(X_1, X_2, \dots\) v-a i.i.d tal que \(P(X_i = 1) = p \land P(X_i = -1) = q\) Supongamos que \(X_0 = 0\) y sea \(\psi_n = \sum_{i = 0}^nX_i\) luego \[\mathbb{E}[\psi_n] = \mathbb{E}\left[\sum_{i = 0}^nX_i\right] = \sum_{i=0}^n \mathbb{E}[X_i] = \sum_{i=0}^n p - q = n(p - q)\]

Ahora la varianza de \(\psi_n\) \[ var[\psi_n] = var\left[\sum_{n=1}^n X_i\right] = \sum_{n=1}^n var[X_i] = n \cdot var[X_i] = n\cdot[\mathbb{E}[X^2] - \mathbb{E}[X] \] \[= n\cdot(p + q - (p - q)^2) = n\cdot(1 - p^2 - 2pq + q^2) = n\cdot(1 - p^2 + 2pq - q^2) \] \[= n\cdot(1 - p^2 + 2pq - q^2 + 2pq - 2pq) = n\cdot(1 - 4pq - (p + q)^2) = 4npq\]

# Simulacion de caminata aleatoria 
library(ggplot2)
for(i in 1:1000){
  p <- 0.5
  n <- 500
  U <- runif(n)
  y <- cumsum((U <= p) - (U > p)) - y[1]
  Z[i] <- y[n]
}
qplot(Z, geom="histogram", binwidth=2)

Calculemos: \[ P(\psi_n = x) \] Sea \[ I_n = \text{numero de moviminetos a la izquierda} \] \[ D_n = \text{numero de moviminetos a la derecha} \] \[ I_n + D_n = n \quad D_n = \sum \frac{X_i + 1}{2} = \frac{\sum X_i}{2} + \frac{n}{2}\] \[ P(\psi_n = x) = P\left(D_n = \frac{x + n}{2}\right) = \binom{n}{\frac{x+n}{2}} p^{\frac{x+n}{2}} q^{\frac{n-x}{2}}\]

Probabilidad de volver a pasar por el punto inicial

\[ 1 - | p - q | = \begin{cases} 1 & p = q\\ < 1 & p\neq q\end{cases} \]

Caminata aleatoria bidimensional

# Parameters
n <- 10
steps <- 1000
left <- 0.5
up <- 0.5
# Simulation
route <- matrix(ncol = 2)
for(x in 1:n){
  walker <- matrix(c(0,0), nrow = steps+1, ncol = 2, byrow = T)
  for(i in 1:steps){
    hor <- rbinom(1, 1, left)
    if(!hor){
      walker[i+1,1] <- walker[i,1] + 1
    }
    if(hor){
      walker[i+1,1] <- walker[i,1] - 1
    }
    ver <- rbinom(1, 1, up)
    if(ver){
      walker[i+1,2] <- walker[i,2] + 1
    }
    if(!ver){
      walker[i+1,2] <- walker[i,2] - 1
    }
  }
  route <- rbind(route, walker)
}
id <- rep(1:n, each = 1001)
colnames(route) <- c("x" , "y")
route <- as.data.frame(route)
route <- cbind(route[2:nrow(route),], org = factor(id))
# Plot
p <- ggplot(route, aes(x = x, y = y, colour = org))
p <- p + geom_path()
print(p)