Biased random walk ($\mathbb{E}{S}_n=n(2p-1),p\ne \frac{1}{2}$)

A biased random walk on the integer line takes unit steps right and left with unequal probabilities $p$ and $1-p$, using independent and identically distributed random variables as increments. For $S_0=0$, the expected value is $n(2p-1)$: positive for $p>1/2$ and negative for $p<1/2$. Changing $p$ to $1-p$ reverses the preferred direction. The exact endpoint probability mass function is given under simple random walk on the integer line.

For example, with $p=0.65$ each step has expected value $0.30$ and variance $0.91$. After $80$ steps,The displacement of the expected value from zero is about $24/8.532\approx 2.813$ standard deviations, yet individual sample paths still move left as well as right, as the figure under simple random walk illustrates. Bias lets this elementary stochastic process model an average directional tendency while retaining fluctuations. Boundary and return questions have their own treatments under gambler's ruin and the specialized children below.

Let $S_0=0$ and let the independent and identically distributed random variables $S_n-S_{n-1}$ be $+1$ with probability $p>1/2$ and $-1$ with probability $q=1-p>0$. The strong law of large numbers gives $S_n\to +\infty$ almost surely, so the global minimum $W$ is finite and attained. For each nonnegative integer $m$, the unit-step property and translation of the starting point giveby the biased random walk hitting probability. Subtracting consecutive tails gives the displayed zero-based geometric distribution, with success parameter $1-q/p$. Inclusion of time zero is important: it ensures $-W\ge 0$. Reflecting the sign of the path yields the corresponding maximum of a downward-biased random walk.

For a biased random walk on the nonnegative integers with upward probability $p>q=1-p>0$, the hitting probability of zero from $i$ is $(q/p{)}^i$. Indeed, first-step analysis gives $h_i=p{h}_{i+1}+q{h}_{i-1}$ for $i\ge 1$, with $h_0=1$. Its characteristic roots are $1$ and $\beta =q/p$, so any solution is $A+(1-A){\beta }^i$. Nonnegativity as $i\to \infty$ forces $A\ge 0$ and hence a value at least ${\beta }^i$. This candidate itself satisfies the recurrence. The theorem on the minimal nonnegative solution for hitting probabilities therefore identifies it with $h_i$. A walk on all of $\mathbb{Z}$ has the same probability until it first reaches zero.

For a walk with increments $+1$ with probability $p$ and $-1$ with probability $1-p$, putThen $\exp (\theta S_n-n\psi (\theta ))$ is a martingale. Optional stopping at the first linear-boundary crossing and optimization over $\theta$ gives exponential maximal-deviation bounds governed by the Legendre transform of a cumulant-generating function.

On $\{0,\dots ,N\}$, stop at zero, move left with probability $q>1/2$ and right with probability $p=1-q$ at interior states, and move deterministically from $N$ to $N-1$. For $m_i={\mathbb{E}}_i{T}_0$, the first-step analysis gives $m_0=0$, $m_i=1+q{m}_{i-1}+p{m}_{i+1}$ and $m_N=1+m_{N-1}$. Setting $D_i=m_i-m_{i-1}$ gives $q{D}_i-p{D}_{i+1}=1$, $D_N=1$, hence $D_i=[1-2p(p/q{)}^{N-i}]/(q-p)$. Summing these differences proves the displayed formula, and more generally $m_i=i/(q-p)-2pq(p/q{)}^{N-i}[1-(p/q{)}^i]/(q-p{)}^2$. A downward block has a uniformly positive probability of reaching zero, which supplies a geometric tail bound and justifies finiteness of these expectations.

For independent lattice steps of length $a$ at intervals $\tau$, with probabilities $\alpha +\beta =1$, the diffusive scaling $a^2/\tau \to 2D$ and $(\alpha -\beta )a/\tau \to V$ requires a bias of order $a$. Expanding the master equation gives the advection-diffusion equation $p_t+V{p}_x=D{p}_{xx}$. The finite-step variance coefficient is $2a^2\alpha \beta /\tau$; it has the same limit because the squared bias vanishes. Fixed nonzero bias does not give finite drift under this diffusive scaling.

For a nearest-neighbour random walk with upward probability $0<p<q=1-p$, start at $S_0=0$ and put $M={\sup }_{n\ge 0}{S}_n$. Its probability distribution is geometric on the nonnegative integers, with the displayed mass and expected value $p/(q-p)$. To prove this, the exponential martingale of a biased simple random walk $Z_n=(q/p{)}^{S_n}$ has expectation one. Stop at the first hit $H_k$ of $k$. Before that hit, $Z_{n\wedge H_k}\le (q/p{)}^k$; the strong law of large numbers gives $S_n\to -\infty$ on paths without a hit. Dominated convergence theorem therefore gives $1=(q/p{)}^k\mathbb{P}(H_k<\infty )$. Taking differences of $\mathbb{P}(M\ge k)=(p/q{)}^k$ yields the mass. If time zero is excluded, the maximum can equal $-1$ with probability $q-p$ and does not have this nonnegative geometric distribution.

Let a discrete-time chain move up with probability $q$ and down with probability $p=1-q$ away from zero, while at zero it moves up with probability $q$ and stays with probability $p$. It is transient for $q>p$, null recurrent for $q=p$, and positive recurrent for $q<p$. In the positive-recurrent case its invariant distribution isand aperiodicity from the self-loop at zero implies convergence to this distribution.