[stationary_densities] Fix typos and align with style manual

[jstac]

Summary

Two commits cleaning up the stationary_densities lecture:

1. Typos and minor code issues

• "for all $s$" → "for all $x$" in the state-space remark
• Off-by-one comment on density dates (1,...,T+1 → 1,...,T)
• Stray trailing comma in the boxplot data display
• "discrete-time" → "discrete time" (noun usage)
• Missing period after the Markov operator definition
• Initialize X[0] in the exercise 1 solution (was uninitialized np.empty memory)
• Boxplot panel titles now label the initial condition as $X_0$, not $t$
• Comments corrected to $B \sim N(0, a_\sigma^2)$ since a_σ is the standard deviation

2. Style-manual alignment

• Links: intersphinx {doc} links for finite_markov references, normalized the outdated python-intro.quantecon.org domain, auto-title link for arma
• Citations: {cite:t} for the in-sentence Hopenhayn–Rogerson reference
• Definitions: bold (not italic) for stochastic kernel, Markov operator, look-ahead, stationary, global stability, ergodicity
• Math: square brackets for expectation, \mathrm{LN} for the lognormal, "CDFs" capitalization
• Figures: ax.set_title → mystnb caption on the main example, caption for the static stability figure, dropped unneeded figsize settings, ex-3 panel titles formatted to one decimal
• Code: migrated to the np.random.default_rng() Generator API, with random_state=rng on all scipy .rvs() calls for reproducibility

Deliberate deviations from the style manual

• Kept figsize=(11, 8) on the 2×2 panel in the ex-2 solution: this repo has no series-wide figure-size default in _config.yml, and at matplotlib defaults the grid's labels collide. (The repo may want the standard figure-size config added separately.)
• Kept the ex-3 subplot titles: mystnb captions break the PDF build inside solution directives, and the titles carry the per-panel initial conditions.
• Kept \mathscr D for the set of densities (plain $D$ would collide visually with other notation).

Verification

Converted with jupytext --to py and executed end-to-end — clean run. Visually inspected all five generated figures: density sequences converge, the look-ahead estimator remains visibly tighter than the kernel estimate, and the ex-2/ex-3 convergence patterns match the live site, so the surrounding narrative still holds. Note figures will differ from the live site in exact draws (PCG64 vs legacy global state) per the RNG migration guidance.

🤖 Generated with Claude Code

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[jstac]

@Chihiro2000GitHub @xuanguang-li If either of you have time could you please review these changes, which should be in line with the manual, fix the build error (ask claude to investigate), and let me know when you are done? If you can't do it please pass it to @mmcky

[github-actions]

🚀 Deployed on https://6a2121c2554c3b2f474e02e6--lustrous-melomakarona-3ee73e.netlify.app

[xuanguang-li]

Thanks, @mmcky, for fixing the compilation error.

I’ll be able to review this lecture in one or two weeks, as I’m currently reviewing the conversion of DP Vol. II.

Maybe we can discuss the schedule and decide how to proceed. @Chihiro2000GitHub

[jstac]

Many thanks @mmcky

[Chihiro2000GitHub]

Hi @jstac, @mmcky, @xuanguang-li,

Sorry for the delayed response.

Many thanks, @mmcky, for fixing this. Glad this has been resolved.

Nice to meet you here, @xuanguang-li.
I hope we'll have another chance to work together sometime.