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arxiv: 2507.15475 · v2 · pith:Y2VKIDZ5new · submitted 2025-07-21 · 📡 eess.SP · math.PR· stat.AP

On the Distribution of a Two-Dimensional Random Walk with Restricted Angles

Karl-Ludwig Besser This is my paper

Pith reviewed 2026-05-22 00:38 UTC · model grok-4.3

classification 📡 eess.SP math.PRstat.AP
keywords random walkrestricted anglesprobability distributionsupport characterizationtwo-dimensionalsignal processing
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The pith

A two-dimensional random walk with each angle restricted to an arbitrary subset of the circle has an exactly characterizable support and closed-form distributions for two steps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines two-dimensional random walks where each step's direction must come from some fixed but arbitrary subset of all possible angles. It gives closed-form expressions for the joint and marginal probability distributions after exactly two steps. For larger numbers of steps it supplies numerical methods to compute the distributions together with approximations that apply when the step count is high. It also supplies a precise geometric description of the set of positions that can be reached at all, no matter how many steps are taken. These results are intended as a reference for problems in signal processing that involve similar angle constraints.

Core claim

We derive the exact joint and marginal distributions for two steps, numerical solutions for a general number of steps, and approximations for a large number of steps. Furthermore, we provide an exact characterization of the support for an arbitrary number of steps.

What carries the argument

The exact characterization of the support as the set of all possible vector sums whose angles lie inside the allowed subset.

If this is right

  • The joint distribution after two steps is obtained by integrating over pairs of allowed angles.
  • Repeated convolution of the single-step distribution produces the position distribution for any fixed number of steps.
  • For large step counts the position distribution approaches a Gaussian whose variance depends on the allowed angles.
  • The reachable region after n steps is the n-fold vector sum of the allowed single-step set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The support description could guide the choice of allowed angle sets to control coverage in wireless aggregation schemes.
  • The same summation and integration approach may extend directly to walks with random step lengths.
  • Numerical methods developed here could be adapted to check higher-dimensional versions of the same angle restriction.

Load-bearing premise

Each step angle is drawn independently from a distribution supported only inside the given subset of the circle.

What would settle it

A Monte Carlo sample of two-step positions that lies outside the analytically predicted support for a chosen angle subset, or a direct numerical check that disagrees with the derived two-step density formula.

Figures

Figures reproduced from arXiv: 2507.15475 by Karl-Ludwig Besser.

Figure 2
Figure 2. Figure 2: PDF of the resulting angle θ2 for a = 0.5. Besides the exact solution from (7), a numerical PDF obtained through MC simulation with 106 samples is shown. (Example 1) Combing (14) with relation (10), we can write the conditional probability as FR2|θ2 (r | θ) = Pr (R2 ≤ r | θ2 = θ) = Pr  cos (2(φ2 − θ)) ≤ r 2 2 − 1 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the event for calculating the conditional probabil [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of the radius R2 after two steps for different values of the maximum angle a. The case of a = π corresponds to the traditional model of having a uniform distribution of the angles over the full circle. (Example 3) A. Characterization of the Support While it is ideal to have an exact characterization of the joint distribution in closed form, it can sometimes be sufficient to only have an exact … view at source ↗
Figure 7
Figure 7. Figure 7: Support and its boundaries Sin and Sout for N = 3 and a = 0.85. Additionally, a section of the circle with the minimum radius Rmin is shown. (Example 4) θN,in(t) = arctan (N − 1 − 2k(t)) sin a + sin φ(t) (N − 1) cos a + cos φ(t) (38) with k(t) = ⌈N t − 1⌉ (39) φ(t) = a [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the relevant slopes of Sin in comparison with the angle θ, determining whether the radius of Sin is a function of the angle. (Lemma 1) 0 1 2 3 4 −4 −2 0 2 4 R4 cos θ4 R4 sin θ4 (a) Support in Cartesian coordi￾nates (real and imaginary parts of Z) −1 0 1 1 2 3 4 Rmin,4 θ4 R4 (b) Support in polar coordinates [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Support and its boundaries for N = 4 and a = 1.4. Additionally, a section of the circle with the minimum radius Rmin is shown. The black dashed line indicates an angle for which more than one point in Sin exists, since condition (41) is violated. (Example 5) Therefore, if the maximum angle a is smaller than this value, condition (41) in Lemma 1 is fulfilled for all N. Example 5 (Non-unique Radius in Sin). … view at source ↗
Figure 10
Figure 10. Figure 10: Distributions of a random walk with N = 3 steps and maximum angle a = 0.5. Besides the numerical solution from Theorem 3, the approxi￾mation for large N from Section IV, and a histogram obtained though MC simulations with 107 samples is shown. (Example 6) Example 6 (Three Steps N = 3). The results of Theorem 3 are illustrated with the example of a three step random walk with maximum angle a = 0.5, for whi… view at source ↗
Figure 11
Figure 11. Figure 11: Approximation of the distribution of the radius [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Approximation of the joint density of the radius [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

In this paper, we derive the distribution of a two-dimensional (complex) random walk in which the angle of each step is restricted to a subset of the circle. This setting appears in various domains, such as in over-the-air computation in signal processing. In particular, we derive the exact joint and marginal distributions for two steps, numerical solutions for a general number of steps, and approximations for a large number of steps. Furthermore, we provide an exact characterization of the support for an arbitrary number of steps. The results in this work provide a reference for future work involving such problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript derives the exact joint and marginal distributions of a two-dimensional random walk whose step angles are restricted to an arbitrary subset of the circle. Exact results are obtained for two steps by direct integration over the allowed angles; numerical convolutions are used for a general number of steps; large-step approximations and an exact geometric characterization of the reachable support are also provided. The work is motivated by applications such as over-the-air computation in signal processing.

Significance. If the derivations are correct, the closed-form two-step distributions and the support characterization supply useful reference results for constrained random walks. These exact elements are independent of fitted parameters and rest on direct integration and vector-addition geometry, which strengthens their value as a foundation for future analytic work in signal processing.

major comments (1)
  1. [Numerical results / general-step section] The numerical convolution procedure presented for a general number of steps does not include error bounds, convergence rates, or validation against the exact two-step case, which limits the ability to assess the accuracy of the reported distributions for n>2.
minor comments (1)
  1. [Abstract] The abstract states that numerical solutions are provided but does not indicate the discretization method or any accuracy metric used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We address the single major comment below and have incorporated revisions to strengthen the numerical section.

read point-by-point responses

  1. Referee: [Numerical results / general-step section] The numerical convolution procedure presented for a general number of steps does not include error bounds, convergence rates, or validation against the exact two-step case, which limits the ability to assess the accuracy of the reported distributions for n>2.

    Authors: We agree that explicit validation and error analysis would improve the presentation of the numerical results. In the revised manuscript we have added a dedicated paragraph and accompanying figure that directly compares the numerical convolution output for n=2 against the exact closed-form joint and marginal distributions derived in Section III. We also include a short discussion of the discretization error arising from the angular quadrature, together with a conservative a-priori bound on the L1 error that scales with the angular step size; this bound is used to report practical error estimates for the n>2 cases shown in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper's core results follow from direct integration of the joint density over independent angles drawn from a fixed subset, followed by convolution for additional steps and a geometric description of the reachable support set. These operations are standard probabilistic constructions that start from the stated independence and uniformity assumptions inside the subset and produce the claimed joint/marginal densities and support characterization without reducing any output to a fitted parameter, self-citation, or redefinition of the input. No load-bearing step equates a derived quantity to its own defining equation or to prior work by the same author.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rest on standard properties of complex addition and independent angle selection; no new physical entities or ad-hoc constants are introduced. The only modeling choice is the fixed subset of allowed angles, treated as an input rather than a fitted parameter.

axioms (2)
  • domain assumption Step angles are chosen independently according to a distribution supported on a fixed subset of the circle.
    Stated in the problem setup; required for the convolution and integration steps to be well-defined.
  • domain assumption Step lengths are independent of angles and of each other.
    Implicit in the two-dimensional random-walk model; allows separation of magnitude and phase in the complex-plane representation.

pith-pipeline@v0.9.0 · 5617 in / 1520 out tokens · 42583 ms · 2026-05-22T00:38:06.695780+00:00 · methodology

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Reference graph

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