M.9.2 Ensemble vs Time Averages

Basic Prerequisites: 01 · From Zero.

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1. Intuition & Practical Objective

Page 01 showed the paradox on a coin. This page names the machinery: what ergodicity is, why additive processes have it, why multiplicative processes lose it, and how to tell the two apart before you are ruined by one.

The practical objective is a test you can apply: given a quantity that evolves, is its ensemble average a fair description of what one participant experiences over time? For a sum (additive) the answer is yes - the law of large numbers makes one long path’s average converge to the group mean. For a product (multiplicative, i.e. any compounding account) the answer is no - the group mean is inflated by a vanishing tail, and the typical path is governed by a different, smaller number.

Two vocabulary items make the rest of the folder precise:

• Ensemble average $⟨x⟩$: freeze time, average over many parallel worlds. This is what a backtest across many assets or a Monte Carlo expectation computes.
• Time average $\bar{x}$: freeze the world (one realisation), average over a long stretch of time. This is what your account statement computes.

A process is ergodic when the two coincide almost surely. Wealth is not ergodic. That single fact is the reason a fund that looks superb in a cross-sectional study can blow up the one account that actually holds the position.

Takeaway. Ergodicity is a property you must check, not assume. Additive increments buy it; multiplicative compounding does not.

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2. Mathematical Ground Truth & Derivations

2.1 The definitions

For a stochastic process $x(t)$:

$$⟨x(t)⟩=\int xd\mathbb{P}(x)\text{(ensemble)},\overline{x}=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_0^{T}x(t)dt\text{(time)}.$$

Ergodic (for the mean): $⟨x(t)⟩=\overline{x}$ almost surely. Birkhoff’s ergodic theorem guarantees the time average exists and equals the ensemble average when the process is stationary and ergodic.

2.2 Additive processes are ergodic

Let $X_i$ be i.i.d. with mean $\mu <\infty$. Then $S_N={\sum }_{i=1}^N{X}_i$ satisfies, by the strong law of large numbers,

$$\overline{X}=\underset{N\to \infty }{\lim }\frac{S_N}{N}=\mathbb{E}[X_i]=\mu \text{a.s.}$$

The time average of the increments equals the ensemble mean of the increments. A random walk’s level wanders (its own time average may not converge), but the increment rate is ergodic. Additive accumulation has no ensemble/time split in the growth rate.

2.3 Multiplicative processes are not ergodic

Let returns $R_i$ be i.i.d. and $W_N=W_0{\prod }_{i=1}^N(1+R_i)$. The two “average growth rates” differ:

$$\underset{\text{time-average (typical) growth}}{\underbrace{\frac{1}{N}\ln \frac{W_N}{W_0}\stackrel{\text{a.s.}}{\to }\mathbb{E}[\ln (1+R)]=g}},\underset{\text{ensemble growth}}{\underbrace{\frac{1}{N}\ln \mathbb{E}\left[\frac{W_N}{W_0}\right]=\ln \mathbb{E}[1+R]}}.$$

By Jensen’s inequality $\mathbb{E}[\ln (1+R)]\le \ln \mathbb{E}[1+R]$, with strict inequality whenever $R$ is non-degenerate. So:

$$\boxed{\text{ensemble growth rate}\ge \text{time-average growth rate}}$$

and the gap is precisely the variance penalty of page 03. When $\mathbb{E}[1+R]>1$ but $\mathbb{E}[\ln (1+R)]<0$ the process is positively expected yet almost surely decaying - the coin game of page 01, and the object lesson of non-ergodicity.

Why the ensemble is misleading here. $\mathbb{E}[W_N]=W_0\mathbb{E}[1+R{]}^N$ is carried by exponentially rare paths: the mean is not a “typical” value. The median is the typical value, and it tracks the time average: $\mathrm{med}(W_N/W_0)\approx e^{Ng}$ for large $N$. A distribution whose mean and median diverge exponentially is the definition of a non-ergodic wealth process.

2.4 The additive ↔ multiplicative dictionary

Feature	Additive	Multiplicative
State update	$x_{t+1}=x_t+X_t$	$W_{t+1}=W_t(1+R_t)$
Natural scale	level $x$	log level $\ln W$
Ergodic in rate?	yes	no
Typical growth	$\mathbb{E}[X]$	$\mathbb{E}[\ln (1+R)]$
Ensemble growth	$\mathbb{E}[X]$	$\ln \mathbb{E}[1+R]$
Fat-tail danger	none	extinction, ruin

The moral: anything that compounds (capital, information, population) must be analysed in logs, where it becomes additive and ergodic again. That is why every growth formula in this folder is written for $\ln W$.

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3. Computational Implementation - the two games side by side

Stdlib only. We run the same two-sided coin two ways - once additively, once multiplicatively - and watch the time average and ensemble average agree in one case and split in the other.

The additive path’s time average ($0.00501$) sits on its ensemble mean ($0.00500$): ergodic. The multiplicative game’s time average ($-0.053$ per round, matching the theoretical $-0.0527$) has the opposite sign to its ensemble growth ($+0.0488$): non-ergodic, with the divergence growing without bound.

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4. Failure Modes & First-Principles Breakdowns

1. Using the ensemble mean as a forecast for one path. The dollar expectation is dominated by paths that will never be yours. Report the median (or the time-average growth) when the decision is about a single account.
2. Reading a positive backtest mean as a positive growth rate. A cross-sectional average of returns is an ensemble object; a single strategy’s compounded equity is a time object. They diverge by exactly $\ln \mathbb{E}[1+R]-\mathbb{E}[\ln (1+R)]$.
3. Forgetting that the ensemble mean only exists because of divergence. $\mathbb{E}[W_N]\to \infty$ while $W_N\to 0$ a.s. is possible; the mean is not “wrong”, it is non-representative. Both limits can be true of the same process.
4. Assuming stationarity. All of this presumes the return distribution is time-invariant. Real markets drift ($\mathbb{E}[1+R]$ and $\mathrm{Var}$ change); the “time average” then only converges to a time-varying ensemble, which is why page 06 leans so hard on estimation error.

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5. References

• Peters, Ole: The Ergodicity Problem in Economics, Nature Physics 15, 1216–1221 (2019)
• Peters & Gell-Mann: Evaluating Gambles Using Dynamics, Chaos 26, 023103 (2016)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• Shreve, Steven E.: Stochastic Calculus for Finance I

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Multiplicative Growth · Index Hub
• Theory: Probability & Measure Theory (LLN, Jensen) · Brownian Motion & Martingales
• Applications: 04 · Kelly Criterion · Econometrics & Time Series