Skip to content

Risk Modelling and Survival Analysis · Core concepts of time series models

White Noise and Random Walk Processes in Time Series

Updated 11 October 2026 · Fact-checked

White noise is a sequence of uncorrelated random variables with mean zero and constant variance σ². A random walk is the running sum of white noise: Xt = Xt-1 + et. It is not stationary because its variance grows with t. Differencing a random walk gives back white noise.

Understand White Noise and Random Walk Processes

White noise is the simplest time series. Each value et has mean 0 and variance σ². Values at different times are uncorrelated. That is all the definition needs. Many exam questions also say the et are independent and normal, but that is an extra assumption, not part of the basic definition.

Why does it matter? White noise is the building block. AR, MA and ARMA models are all built by feeding white noise through a rule. White noise is also what you hope to see in the residuals of a fitted model. If the residuals are white noise, the model has captured the structure in the data.

A random walk adds up white noise. Start at X0 = 0 and set Xt = Xt-1 + et. Then Xt = e1 + e2 + ... + et. Each step is a random shock, and the shocks pile up. The process never forgets a shock, so it wanders and does not return to a fixed level.

Because of this, the random walk is non-stationary. Its mean may be constant (zero), but its variance is tσ², which grows with time. Its covariance also depends on time, not only on the lag. A random walk with drift adds a constant μ each step: Xt = Xt-1 + μ + et. Now the mean grows linearly too.

The link between the two is differencing. Take Xt - Xt-1 and a random walk becomes white noise (or white noise plus a constant if there is drift). This is why a random walk is called an integrated process of order 1, or I(1). It is the simplest ARIMA(0,1,0) model.

Key rules to remember

White noise definition
E[et] = 0, Var(et) = σ², Cov(et, es) = 0 for t ≠ s
Uncorrelated is enough. Independence is a stronger, separate assumption.
White noise autocorrelation
ρk = 1 for k = 0, and ρk = 0 for k ≥ 1
Sample autocorrelations of white noise should lie within about ±2/√n.
Random walk
Xt = Xt-1 + et, so Xt = X0 + e1 + ... + et
With X0 fixed (often 0).
Random walk mean and variance
E[Xt] = X0, Var(Xt) = tσ²
Assumes X0 is a constant. Variance grows with t, so not stationary.
Random walk autocovariance
Cov(Xt, Xt+k) = tσ² for k ≥ 0
Depends on t, not just the lag k.
Random walk with drift
Xt = Xt-1 + μ + et, E[Xt] = X0 + μt, Var(Xt) = tσ²
Drift changes the mean only. The variance is unchanged.
Differencing
∇Xt = Xt - Xt-1 = μ + et
Result is stationary: white noise with mean μ.

How to solve White Noise and Random Walk Processes questions

Use this method for any question on white noise or random walks, whether it asks for moments, stationarity or a forecast.

  1. 1Write the model clearly. Identify whether it is white noise, a random walk, or a random walk with drift, and note the starting value X0.
  2. 2State the assumptions on et: mean 0, variance σ², uncorrelated (or independent, or normal, if given).
  3. 3Expand the process as a sum: Xt = X0 + μt + e1 + ... + et.
  4. 4Take the expectation term by term to get the mean. Constants stay, the et terms drop out.
  5. 5Take the variance. Use uncorrelated shocks so variances add: Var(Xt) = tσ². A constant X0 or μt adds nothing.
  6. 6For covariance, count the shared shocks. Cov(Xs, Xt) = min(s, t)σ².
  7. 7Decide on stationarity. If the mean or variance depends on t, the process is not stationary. State this and give the reason.
  8. 8If asked, difference once to reach a stationary series, and state the mean and variance of the result.

Quickest way: Count the shocks

When to use it: Use for moment questions on a random walk when time is short.

  1. Write Xt as X0 + μt plus t shocks.
  2. Mean = X0 + μt. Variance = t × σ².
  3. Covariance between times s and t = number of shared shocks × σ² = min(s, t)σ².
  4. Check: if the answer depends on t, it is not stationary. Differencing leaves one shock, so the variance is σ².

Common mistakes in White Noise and Random Walk Processes

  • Saying a random walk has variance σ² because the shocks have variance σ².

    You confuse the shock et with the level Xt.

    Fix: Xt is a sum of t shocks. Variance is tσ². Only the first difference has variance σ².

  • Adding drift into the variance, for example Var(Xt) = tσ² + μt.

    You treat μt like a random term.

    Fix: Drift is a constant. It shifts the mean and has zero variance.

  • Calling a random walk stationary because its mean is constant.

    Weak stationarity needs three conditions, and you check only one.

    Fix: Check mean, variance and covariance. The variance tσ² grows with t, so it fails.

  • Assuming white noise must be normal or independent.

    Most examples use normal shocks.

    Fix: White noise needs only zero mean, constant variance and zero correlation. Say normal or independent only if the question states it.

  • Writing Cov(Xs, Xt) = σ² min(s, t) but applying it with the wrong starting point, or ignoring that X0 is random.

    You assume X0 = 0 without checking.

    Fix: If X0 is a constant, it adds nothing. If it is random, add its variance and covariance terms explicitly.

Worked examples

Example 1

Let Xt = Xt-1 + 0.5 + et, with X0 = 10 and et white noise with variance 4. Find E[X20] and Var(X20), and state whether the process is stationary.

Show the solution
  1. The model is a random walk with drift μ = 0.5, and X0 = 10 is a constant.
  2. Expand: X20 = 10 + 20 × 0.5 + e1 + ... + e20 = 20 + (e1 + ... + e20).
  3. Mean: E[X20] = 20, since each et has mean 0.
  4. Variance: shocks are uncorrelated, so Var(X20) = 20 × 4 = 80.
  5. Both the mean 10 + 0.5t and the variance 4t depend on t, so the process is not stationary.

Answer: E[X20] = 20, Var(X20) = 80. The process is not stationary because its mean and variance change with t.

Example 2

For a random walk Xt = Xt-1 + et with X0 = 0 and Var(et) = σ² = 9, find Cov(X4, X10) and Corr(X4, X10). Then give the distribution of the first difference.

Show the solution
  1. Write X4 = e1 + e2 + e3 + e4 and X10 = e1 + ... + e10.
  2. Shared shocks are e1 to e4, so Cov(X4, X10) = 4 × 9 = 36.
  3. Var(X4) = 4 × 9 = 36 and Var(X10) = 10 × 9 = 90.
  4. Corr = 36 ÷ √(36 × 90) = 36 ÷ √3240 = 36 ÷ 56.92 = 0.632.
  5. Check: this equals √(4/10) = √0.4 = 0.632.
  6. First difference: ∇Xt = Xt - Xt-1 = et, which is white noise with mean 0 and variance 9.

Answer: Cov(X4, X10) = 36, Corr ≈ 0.632. The first difference is white noise with mean 0 and variance 9.

Exam tips

  • Always state the stationarity conclusion with its reason. Marks are usually given for naming which condition fails.
  • In written answers, show the expansion Xt = X0 + μt + e1 + ... + et before taking moments.
  • For MCQs, test the options quickly: mean X0 + μt and variance tσ² are the two values most often examined.
  • If a question asks for residual checks, say that white noise residuals should show sample autocorrelations near zero, within about ±2/√n.
  • In the computer-based paper, simulate a random walk with a cumulative sum of random normals. Difference it and check the plot and ACF.

Practice questions from Core concepts of time series models

White Noise and Random Walk Processes in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

White Noise and Random Walk Processes: frequently asked questions

What is the difference between white noise and a random walk?

White noise is a sequence of uncorrelated shocks with constant mean and variance, and it is stationary. A random walk is the cumulative sum of those shocks, so its variance grows with time and it is not stationary. Differencing a random walk returns white noise.

Is white noise always independent?

No. White noise requires zero mean, constant variance and zero correlation between different times. Independence is stronger. For normal shocks the two ideas coincide, but in general uncorrelated does not mean independent.

What are the mean and variance of a random walk with drift?

With a constant start X0, the mean is X0 + μt and the variance is tσ². Drift changes only the mean. Both depend on t, so the process is non-stationary.

How do I make a random walk stationary?

Take the first difference, Xt - Xt-1. For a random walk this gives white noise. With drift μ it gives white noise with mean μ. This is why a random walk is an ARIMA(0,1,0) process.