8  Stochastic processes

Author

Affiliation

Aditya Mahajan

McGill University

1. A stochastic process is a family of random variables, indexed by time. You may think of it as a random signal.

2. A stochastic process may be described in discrete-time, where we have a family of random variables \(\{X_n : n \in N\}\) where index set \(N\) is either \(\{0, 1, \dots, \}\) or \(\{1, 2, \dots\}\) or in some cases \(\integers\).

3. Note that all random variables must be defined on a common probability space \((Ω, \ALPHABET F, \PR)\). For a fixed \(ω \in Ω\), we get a sequence of real numbers \(\{X_n(ω)\}_{n \ge 0}\). Such a sequence is called a realization or sample path.

4. As an example, consider an infinite sequence of bits, where each bit is i.i.d. \(\text{Bernoulli}(p)\). A sample path is shown below.

viewof rerun_bits= Inputs.button("Generate sample path")

points_bits = {

    rerun_bits

    const N = 120
    const p = 0.5
    var points = new Array(N)

    for(var n=0; n < N; n++) {
      var val = (Math.random() <= p) ? 0 : 1
      points[n-1] = { n: n, X_n: val }
    }
    return points
}

Plot.plot({
  grid: true,
  y : {ticks: 1 },
  height: 100,
  marks: [
    // Axes
    Plot.ruleX([0]),
    Plot.ruleY([0]),
    // Data
    Plot.ruleX(points_bits, {x: "n", y: "X_n", strokeWidth: 1}),
    Plot.dot(points_bits, {x: "n", y: "X_n", fill: "currentColor", r: 2}),
  ]
})

5. As an other example, consider a stochastic process \(\{Y_n\}_{n \ge 0}\) defined as \[ Y_n = x_n + W_n \] where \(\{x_n\}_{n \ge 0}\) is a deterministic discrete-time signal given by \(x_n = \sin(2πn/25)\) and \(\{W_n\}_{n \ge 0}\) is a Gaussian noise process, where \(W_n \sim \mathcal{N}(0,0.2)\). A sample path of \(\{Y_n\}_{n \ge 0}\) is shown below.

function gaussianRandom(mean=0, stdev=1) {
    const u = 1 - Math.random(); // Converting [0,1) to (0,1]
    const v = Math.random();
    const z = Math.sqrt( -2.0 * Math.log( u ) ) * Math.cos( 2.0 * Math.PI * v );
    // Transform to the desired mean and standard deviation:
    return z * stdev + mean;
}

viewof rerun_noise = Inputs.button("Generate sample path")

points_noise = {

    rerun_noise

    const N = 120
    const sigma = 0.2
    const omega = 2*Math.PI*0.04
    var points = new Array(N)

    for(var n=0; n < N; n++) {
      var signal = Math.sin(omega*n)
      var noise = gaussianRandom(0, sigma)

      points[n-1] = { n: n, X_n: signal,  W_n: noise, Y_n: signal + noise }
    }
    return points
}

Plot.plot({
  grid: true,
  y : {ticks: 1 },
  height: 100,
  marks: [
    // Axes
    Plot.ruleX([0]),
    Plot.ruleY([0]),
    // Data
    Plot.ruleX(points_noise, {x: "n", y: "Y_n", strokeWidth: 1}),
    Plot.dot(points_noise, {x: "n", y: "Y_n", fill: "currentColor", r: 2}),
  ]
})

6. It is also possible for a stochastic process to be continuous time, where we have an uncountable collection of random variables \(\{X_t : t \in T \}\), where the index set \(T\) is either \((-∞, ∞)\) or \([0, ∞)\). Continuous-time stochastic processes tend to be a bit more technical than discrete-time stochastic processes.

7. As the above examples illustrate, the random variables \(\{X_n\}_{n \ge 0}\) need not be independent. So, we need a way to specify the joint distribution of a countable collection of random variables. This means that for any collection of time indices \(\{n_1, \dots, n_k\}\), we must be able to specify the joint CDF of the random variables \((X_{n_1}, X_{n_2}, \dots, X_{n_k})\).

8. A discrete time stochastic process \(\{X_n\}_{n \ge 0}\) is called strong-sense stationary if for any \(\{n_1, \dots, n_k\}\) and \(m > 0\), the random vectors \[ (X_{n_1}, X_{n_2}, \dots, X_{n_k}) \quad \text{and}\quad (X_{n_1 + m}, X_{n_2 + m}, \dots, X_{n_k + m}) \] have the same joint distributions.