# Accumulation Rate of Variance in Random Walk

I am slightly confused with the terminology Shreve (2008), he states:

"The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over any time interval $$k$$ to $$l$$ for nonnegative integerst $$k is $$l-k$$."

I understand the latter half of the statement, but I don't understand the variance's accumulation rate. This is something not familar to me. The way I thought of variance's unit is that it is squared, so it is usually not easily interpreted.

The context in which this symmetric random walk is defined is think of a random variable that pays you \$1 if a coin comes up heads and -\$1 otherwise. Consider a cumulative gain or loss of this dollar bet at $$k$$-th bet, and we call this process a symmetric random walk.

So what does he mean here about the rate at which the variance accumulates? The rate at which the dispersion of the cumulative bet accumulates? Confused.

Reference:
Shreve, Steven E. \textit{Stochastic Calculus for Finance II : Continuous-Time Models}. Springer, 2008.

If we denote the random walk with $$(X_k)_{k \in \mathbb{N}}$$ than for all $$k$$ the random variable $$\Delta X_k := X_{k} - X_{k-1}$$ has mean zero and variance one: \begin{align} \mathbb{E}[\Delta X_k] = \frac 12\cdot 1 + \frac 12 \cdot (-1) = 0, \quad \text{Var}(\Delta X_k) = \mathbb{E}[(\Delta X_k)^2] - \bigl(\mathbb{E}[\Delta X_k]\bigr)^2 = 1 \end{align}

This is what is meant by the second part.

The first part deals with the variance of $$X_k$$ which can be computed like this: \begin{align} \text{Var}(X_k) &= \mathbb{E}\Bigl[\text{Var}(X_k \mid X_{k-1}) \Bigr] + \text{Var}\Bigl(\mathbb{E}[X_k \mid X_{k-1}] \Bigr) = 1 + \text{Var}(X_{k-1}), \\ \text{Var}(X_1) &= \mathbb{E}[(X_1)^2] - \bigl(\mathbb{E}[X_1]\bigr)^2 =1. \end{align}

Therefore $$\text{Var}(X_k) = k$$ for all $$k \in \mathbb{N}$$. In particular, the variance of the symmetric random walk accumulates at rate one per unit time.

• Cettt, thanks for the answer. Now, I understand this argument, but my main question is about the interpretation of accumulation at rate one per unit time. Where is the "time" component here. Is the subscript $k$ suppose to represent an index for time? As the dollar bet progresses in time, the dispersion of the cumulative gain increases linearly with respect to time? Is this another way to put it? Nov 19, 2019 at 15:44
• @FrankSwanton, yes exactly. In the application of stochastic processes the index $k$ is often interpreted as time. In your coin flipping example you could say that every minute a coin is flipped and $\Delta X_k$ is the gain or loss at time $k$. Nov 19, 2019 at 15:49
• Right, however, the variance of the symmetric random walk and the variance of the increment of the symmetric random walk are distinct objects. So, when Shreve states the variance of the symmetric random walk, which means the dispersion of the cumulative gain on the dollar bet when you stand in time $k$, accumulates at rate one per unit time, now I understand as that it is the overall cumulative gain's dispersion, not the increment of the gain or loss from one $t$ to $t+1$, is this correct? Nov 19, 2019 at 16:13
• @FrankSwanton your understanding is correct Nov 19, 2019 at 17:55