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Proof that First Differencing Transforms a Random Walk into a Stationary Process

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The Formal Theorem

Let {Xt}t=0∞ \{X_t\}_{t=0}^\infty be a simple random walk, defined by Xt=∑i=1tϵi X_t = \sum_{i=1}^t \epsilon_i for t≥1 t \ge 1 , where ϵi \epsilon_i are independent and identically distributed random variables with E[ϵi]=0 E[\epsilon_i] = 0 and Var(ϵi)=σ2<∞ Var(\epsilon_i) = \sigma^2 < \infty . Let ΔXt=Xt−Xt−1 \Delta X_t = X_t - X_{t-1} for t≥1 t \ge 1 . Then {ΔXt}t=1∞ \{\Delta X_t\}_{t=1}^\infty is a stationary process.

Analytical Intuition.

Imagine a drunkard stumbling randomly on a straight line. Their position at any time t t , Xt X_t , is the sum of all their small, random steps. This path is unpredictable, its mean shifting with time – a non-stationary drunkard! Now, consider the *change* in their position from one second to the next, ΔXt=Xt−Xt−1 \Delta X_t = X_t - X_{t-1} . This change is precisely their single, latest stumble, ϵt \epsilon_t . Since each stumble ϵt \epsilon_t is drawn from the same distribution (same expected size, same variance, no memory of past stumbles), the sequence of stumbles itself possesses constant statistical properties over time. The drunkard's *movement* becomes predictable in its statistical behavior, even if their actual position remains erratic.
CAUTION

Institutional Warning.

Students sometimes confuse the process Xt X_t itself with its increments. The random walk's position accumulates past shocks, leading to time-varying mean and variance. The first difference, however, isolates the current shock, whose distribution is constant.

Academic Inquiries.

01

What is a random walk?

A random walk is a stochastic process that describes a path consisting of a succession of random steps. Mathematically, if Xt X_t is the position at time t t , then Xt=Xt−1+ϵt X_t = X_{t-1} + \epsilon_t , where ϵt \epsilon_t is a random shock at time t t . If the shocks have a mean of zero and are independent, then Xt=∑i=1tϵi X_t = \sum_{i=1}^t \epsilon_i (assuming X0=0 X_0 = 0 ).

02

What does it mean for a process to be stationary?

A stationary process has statistical properties (like mean and variance) that do not change over time. Formally, a process Yt Y_t is strictly stationary if the joint distribution of (Yt,Yt+1,…,Yt+k) (Y_t, Y_{t+1}, \dots, Y_{t+k}) is the same for all t t . A weaker form, covariance stationarity, requires that E[Yt] E[Y_t] is constant, Var(Yt) Var(Y_t) is constant, and Cov(Yt,Yt+h) Cov(Y_t, Y_{t+h}) depends only on the lag h h , not on t t .

03

Why is a random walk *not* stationary?

A standard random walk Xt=∑i=1tϵi X_t = \sum_{i=1}^t \epsilon_i is not stationary because its mean and variance change with time. Assuming E[ϵi]=0 E[\epsilon_i] = 0 and Var(ϵi)=σ2 Var(\epsilon_i) = \sigma^2 , then E[Xt]=E[∑i=1tϵi]=∑i=1tE[ϵi]=0 E[X_t] = E[\sum_{i=1}^t \epsilon_i] = \sum_{i=1}^t E[\epsilon_i] = 0 for all t t . However, Var(Xt)=Var(∑i=1tϵi)=∑i=1tVar(ϵi)=tσ2 Var(X_t) = Var(\sum_{i=1}^t \epsilon_i) = \sum_{i=1}^t Var(\epsilon_i) = t\sigma^2 . Since the variance increases with t t , the process is not stationary.

04

How does first differencing work?

First differencing involves calculating the difference between consecutive observations of a time series. If Yt Y_t is a time series, its first difference is ΔYt=Yt−Yt−1 \Delta Y_t = Y_t - Y_{t-1} . This operation essentially removes the trend or cumulative effect present in the original series.

Standardized References.

  • Definitive Institutional SourceBox, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). *Time Series Analysis: Forecasting and Control*. John Wiley & Sons.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof that First Differencing Transforms a Random Walk into a Stationary Process: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/time-series-analysis/proof-that-first-differencing-transforms-a-random-walk-into-a-stationary-process

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