Biased Simple Random Walk

• Biased Simple Random Walk is a stochastic process on ℤ defined by independent ±1 steps with a bias (p > 1/2) that determines key first-passage properties.
• Explicit generating functions and integral representations yield closed-form expressions for expected win-rate and variance, enabling unbiased Monte Carlo estimators of π.
• Extensions to trees and random environments demonstrate how slight bias induces universal scaling laws, sub-ballistic transport, and Sinai-type slowdowns.

A biased simple random walk is a fundamental stochastic process on $\mathbb{Z}$ (or other graphs) where each step is independently $+1$ (“right”) with probability $p$ and $-1$ (“left”) with probability $q=1-p$, for some $p\in[\tfrac12,1)$. The introduction of bias ($p\ne\tfrac12$) leads to rich phenomena in first-passage problems, empirical frequency limits, scaling laws, and connections to more general structures such as random environments, trees, and random graphs.

1. Definition and Fundamental Properties

Let $(X_n)_{n\ge1}$ be an i.i.d. sequence with $\Pr[X_n=+1]=p$, $\Pr[X_n=-1]=q=1-p$, $p\in[\frac{1}{2},1)$. The walk is defined by $S_0=0$, $S_n=\sum_{k=1}^n X_k$. For a threshold $d\in\mathbb{N}^+$, the first-passage (or hitting) time is $N_d=\inf\{n\ge0: S_n=d\}$, which is almost surely finite if $p\ge\frac{1}{2}$.

The “win count” up to time $n$ is $R_n=\#\{1\leq k\leq n: X_k=+1\}$, and at the stopping time $N_d$, one defines the empirical win-rate

$\hat{p}_{N_d} = \frac{R_{N_d}}{N_d}.$

Utilizing the pathwise identity $S_n=2R_n-n$, it follows that

$$\frac{R_{N_d}}{N_d} = \frac{1}{2} + \frac{d}{2N_d}.$$

These identities provide exact algebraic relationships between the number of “successes” and the random first-passage time.

2. Moment Formulas: Generating-Function and Integral Representations

The analysis of first-passage moments relies on generating functions. Let $T = \inf\{n\ge0: S_n=1\}$ with probability-generating function $$\phi(t) = \mathbb{E}[t^T] = \frac{1-\sqrt{1-4pq\,t^2}}{2qt}$$ for $|t|\le 1$. By translation invariance and the strong Markov property, $N_d$ is the sum of $d$ i.i.d. copies of $T$; hence, $\phi_d(t)=\phi(t)^d$.

The moments of $N_d$ are given by integrals involving $\phi_d(t)$: $$\mathbb{E}\left[\frac{1}{N_d}\right] = \int_0^1 \frac{\phi_d(t)}{t}\,dt,$$

$$\mathbb{E}\left[\frac{1}{N_d^2}\right] = -\int_0^1 \log t\, \frac{\phi_d(t)}{t}\,dt.$$

Adopting the variable change $t = \frac{u}{p+q u^2}$, with $u\in[0,1]$, two key functions are defined: $$h_p(u) = \frac{p-q\,u^2}{p+q\,u^2}, \qquad \ell_p(u) = \log\left(\frac{p+q\,u^2}{u}\right).$$ This leads to fully explicit formulas for the mean and variance of the empirical win-rate at first passage: $$\mathbb{E}\left[\frac{R_{N_d}}{N_d}\right] = \frac{1}{2} + \frac{d}{2} \int_0^1 u^{d-1} h_p(u)\,du,$$

$$\mathrm{Var}\left(\frac{R_{N_d}}{N_d}\right) = \frac{d^2}{4} \left\{ \int_0^1 u^{d-1} h_p(u)\ell_p(u)\,du - \left(\int_0^1 u^{d-1} h_p(u)\,du\right)^2 \right\}.$$

These integral representations are completely explicit up to elementary function evaluation and one-dimensional quadrature (Bruss et al., 24 Dec 2025).

3. Monotonicity, Scaling Limits, and Law of Large Numbers

Systematic variation of $d$ (barrier height) and $p$ (bias) yields rigorous monotonicity and asymptotic behaviors:

• In $d$: The map $$d\mapsto \mathbb{E}\left[\frac{R_{N_d}}{N_d}\right]$$ is strictly decreasing. Explicit calculation shows

$$\mathbb{E}\left[\frac{R_{N_d}}{N_d}\right] > \mathbb{E}\left[\frac{R_{N_{d+1}}}{N_{d+1}}\right] \quad \forall\,d\ge1.$$

One has $$\displaystyle \lim_{d\to\infty} \mathbb{E}[\frac{R_{N_d}}{N_d}]=p$$ and $$\lim_{d\to\infty}\mathrm{Var}[\frac{R_{N_d}}{N_d}]=0$$, so the empirical frequency at first passage converges to the underlying bias (Bruss et al., 24 Dec 2025).

• In $p$: For fixed $d$, the expectation increases strictly with $p$ on $[\frac12,1)$, and as $p\to1$, $\mathbb{E}[R_{N_d}/N_d]\to1$ and $\mathrm{Var}[R_{N_d}/N_d]\to0$. This exemplifies the “vanishing stop-rule bias” at high underlying bias or at large thresholds.

In the unbiased case ($p=\frac12$), $N_d$ exhibits infinite mean and fat tails, while for $p>\frac12$ both mean and variance of $N_d$ are finite and scale linearly in $d$.

4. Closed-Form Solutions for Special Values

For practical computations, explicit closed-form expressions are obtained for $\mathbb{E}[R_{N_d}/N_d]$ at both odd and even threshold values.

Let $d=2m+1$ (odd), $r=q/p$: $$\mathbb{E}\left[\frac{R_{N_{2m+1}}}{N_{2m+1}}\right] = (-1)^m (2m+1) r^{-(m+\tfrac12)} \arctan(\sqrt{r}) + 1 - (2m+1) \sum_{k=0}^m \frac{(-1)^k}{2(m-k)+1} r^{-k}.$$

If $d=2m$ (even): $$\mathbb{E}\left[\frac{R_{N_{2m}}}{N_{2m}}\right] = 1 - \frac{2m}{2\, r^m} \sum_{k=1}^m (-1)^{m-k} \binom{m}{k} \frac{(1+r)^k - 1}{k} + (-1)^{m+1} \frac{m}{r^m} \log(1 + r).$$

In particular, for $p=q=\frac12$ (the fair coin), the odd-$d$ formula yields: $$\mathbb{E}\left[\frac{R_{N_1}}{N_1}\right] = \frac{\pi}{4}, \qquad \mathbb{E}\left[\frac{R_{N_3}}{N_3}\right] = 3\left(1 - \frac{\pi}{4}\right).$$ These explicit finite sum and elementary function formulas allow for both symbolic analysis and high-precision arithmetic evaluation (Bruss et al., 24 Dec 2025).

5. Applications: Unbiased Estimation of $\pi$ via Coin Flipping

The closed-form relationships for $\mathbb{E}[R_{N_d}/N_d]$ directly yield unbiased estimators for transcendental constants. Notably, for the fair coin and odd $d$, the expectation is an affine function of $\pi/4$. By constructing

$$\hat\pi_d = \frac{4}{B_d}\left(\frac{R_{N_d}}{N_d} - A_d\right),$$

with known rationals $A_d, B_d$, one obtains $\mathbb{E}[\hat\pi_d]=\pi$—an exactly unbiased “Monte Carlo” estimator for $\pi$ based purely on coin flips and first-passage sampling. For biased coins (e.g., $p=3/4$), similar affine relationships exploiting $\arctan(\sqrt{q/p})=\pi/k$ allow for unbiased estimators of $\pi$, with the advantage that the stopping time $N_d$ now possesses finite mean and rapidly decaying MSE: $$\text{MSE}[\tilde\pi_d] = O(e^{-cd}), \quad \text{for some } c>0.$$ Numerical evidence supports that high-precision calculations (e.g., 9–12 digits) are achievable with modest $d$ ($\sim10^2$) (Bruss et al., 24 Dec 2025).

A plausible implication is that the interplay between analytic closed-form expectations and engineered stochastic processes enables unbiased simulation-based computation of transcendental constants, with computational efficiency sharply improved by moderating the underlying process bias.

6. Generalizations to Trees and Random Environments

The study of biased simple random walks on more complex structures—Galton–Watson trees and random environments—reveals additional phenomena:

• Galton–Watson Trees: For a $\lambda$-biased random walk on a supercritical Galton–Watson tree, the limiting speed is characterized via an explicit formula,

$$\ell_\lambda = \mathbb{E}_{\mu_\lambda}\left[\frac{\nu - \lambda}{\nu + \lambda}\right],$$

where $\mu_\lambda$ is the invariant measure of the environment seen from the particle, $\nu$ is the root degree, and the formula involves renewal and ergodicity arguments. For the unbiased case ($\lambda=1$), this reduces to $$\ell_1 = \mathbb{E}\left[\frac{\nu-1}{\nu+1}\right]$$ (Aidekon, 2011).

• Random Walk on Random Walk Range: For a biased walk on the range of a simple random walk in $\mathbb{Z}^d$ ($d\ge5$), any nontrivial bias induces a Sinai-type slowdown: $X_n/\log n$ converges in law, with the scaling limit described via the smallest valley of two-sided Brownian motion. Ballistic speed is lost even for small bias, as deep potential wells in the effective one-dimensional RWRE dominate late-time transport (Croydon, 2012).

These results link classical discrete models to contemporary probabilistic, statistical-physical, and simulation applications.

7. Broader Significance and Outstanding Themes

Biased simple random walks provide an analytically tractable setting for the study of first-passage asymptotics, stop-rule biases, and unbiased Monte Carlo estimation of constants. The exact formulas for expectations and variances at stopping times $N_d$ elucidate the bias-variance-computation tradeoff inherent in sequential sampling. Analytical techniques—generating functions, strong Markov property, explicit change of variables—are essential.

The extension to non-homogeneous and high-dimensional environments underscores universal behaviors (Sinai regime, sub-ballistic motion) and exceptional sensitivity of transport properties to bias and structural randomness. The construction of unbiased estimators of $\pi$ via first-passage processes highlights the fusion of probability theory with computational mathematics, with exponential variance decay in the bias parameter.

The persistent theme is that even minimal systematic asymmetry in simple stochastic models can profoundly affect first-passage statistics, limit behavior, and algorithmic efficiency (Bruss et al., 24 Dec 2025, Aidekon, 2011, Croydon, 2012).