I knew the fair-coin answer off the top of my head (probability of ruin is (N-i)/N) but the biased extension tripped me up more than I expected.
Start by setting up the recurrence relation for the probability of ruin, then solve it for the fair coin case using the boundary conditions. For the biased case, solve the resulting linear recurrence with the appropriate characteristic equation, and present the final formula clearly.
Pro tip: Mention that this is a classic martingale problem and that the fair case can be solved instantly by the optional stopping theorem, showing deeper probabilistic insight.
Let P(i) be the probability of ruin starting with i dollars. Set up the recurrence P(i) = p P(i+1) + (1-p) P(i-1) with boundaries P(0)=1, P(N)=0.
For p=0.5, the recurrence simplifies to P(i) = (P(i+1)+P(i-1))/2, which implies P(i) is linear. Apply boundary conditions to get P(i) = 1 - i/N.
Assume solution of form P(i) = r^i. The characteristic equation is p r^2 - r + (1-p) = 0, with roots 1 and (1-p)/p. Use boundary conditions to find constants.
For p≠0.5, P(i) = [ ( (1-p)/p )^i - ( (1-p)/p )^N ] / [ 1 - ( (1-p)/p )^N ]. Simplify and check limits (p→0.5 recovers fair case).
Mention behavior as N→∞, the role of the ratio (1-p)/p, and how this relates to risk of ruin in betting or trading strategies.
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Part (a) I got fine, return probability is 1 in 1D.
Start by defining the simple symmetric random walk and the key quantities: return probability and expected first return time. Use generating functions or combinatorial identities to derive the 1D results, then generalize to higher dimensions using Fourier analysis or Green's functions. Finally, compare the convergence of the return probability series to explain recurrence vs. transience.
Pro tip: Emphasize the difference between recurrence (probability 1 of return) and positive recurrence (finite expected return time). In 1D and 2D, the walk is recurrent but null recurrent (infinite expected return time), while in 3D+ it is transient. This distinction is crucial and often overlooked.
Clearly state the model: simple symmetric random walk on Z^d starting at 0. Define p_d as the probability of ever returning to the origin and E[T] as the expected first return time.
Use the reflection principle or generating functions to show that the probability of return to 0 in 1D is 1. For example, compute the generating function of first return probabilities and evaluate at 1.
Show that the expected return time is infinite by analyzing the generating function or using the fact that the walk is null recurrent. Alternatively, use the known result that E[T] = ∞.
Use Fourier analysis or Green's functions to compute the return probability in d dimensions. Show that the walk is recurrent for d=1,2 and transient for d≥3 by examining the convergence of the integral ∫_{[-π,π]^d} 1/(1 - φ(θ)) dθ, where φ(θ) is the characteristic function.
Summarize: In 1D and 2D, the expected number of returns is infinite, so the walk returns almost surely. In 3D and above, the expected number of returns is finite, so there is a positive probability of never returning.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.