aub_htp.pdf.skorohod#
Functions
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Vectorized dispatcher that combines Skorohod formulas into a stable pdf approximation across parameter regimes. |
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Skorohod's Formula 1: series for 0 < alpha < 1 (tails). |
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Skorohod's Formula 2: alpha = 1 case (Cauchy-like). |
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Helper for Skorohod's Formula 2 (alpha = 1): compute b_k(beta). |
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Skorohod's Formula 3: series for 1 < alpha < 2 (tails). |
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Coefficient a_n(alpha, beta) for Skorohod's Formula 3 (1 < alpha < 2 tails). |
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Skorohod's Formula 4: small-|x| asymptotic for 0 < alpha < 1. |
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Asymptotic prefactor A(alpha) for 0 < alpha < 1 near zero (β = ±1 edges). |
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Asymptotic exponent scale B(alpha) for 0 < alpha < 1 near zero. |
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Lambda(alpha) = alpha / (1 - alpha) for 0 < alpha < 1. |
Skorohod's Formula 5: alpha = 1, near zero correction (heuristic). |
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Skorohod's Formula 6: small-|x| asymptotic for alpha > 1. |
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Asymptotic prefactor A'(alpha) for alpha > 1 near zero (β = ±1 edges). |
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Asymptotic exponent scale B'(alpha) for alpha > 1 near zero. |
Lambda'(alpha) = alpha / (alpha - 1) for alpha > 1. |
- aub_htp.pdf.skorohod.generate_pdf_skorohod_vectorized(x, alpha, beta, epsilon=1e-06)#
Vectorized dispatcher that combines Skorohod formulas into a stable pdf approximation across parameter regimes.
Inputs - x: array of evaluation points (after normalization if needed). - alpha ∈ (0, 2], beta ∈ [-1, 1]. - epsilon: small threshold around zero to switch to near-zero asymptotics.
Logic - alpha < 1:
beta = 1: use Formula 1 on x>ε; Formula 4 near zero on (0, ε].
beta = -1: mirror to left tail; use |x| and flip beta where needed.
else: use Formula 1 on both sides, flipping beta for x<=0.
- alpha = 1:
- beta ∈ [-1,1]:
main: Formula 2 on each side with |x| as needed.
near zero: Formula 5 to stabilize behavior.
- alpha > 1:
beta = 1: Formula 3 on x>0; Formula 6 near/below zero.
beta = -1: swap sides; Formula 6 on x>0; Formula 3 on x<=0 with flip.
else: Formula 3 on both sides, flipping beta for x<0.
Returns - result: array of pdf approximations, same shape as x.
- aub_htp.pdf.skorohod.skorohod_formula_1(x, alpha, beta, N=170)#
Skorohod’s Formula 1: series for 0 < alpha < 1 (tails).
Uses truncated series with N terms.
Valid only when 0 < alpha < 1 and |beta| <= 1.
Works on x > 0 for right tail; use |x| and flipped beta for left tail.
Returns vector of pdf approximations at x.
- aub_htp.pdf.skorohod.skorohod_formula_2(x, beta, N=10)#
Skorohod’s Formula 2: alpha = 1 case (Cauchy-like).
Adjust x by log term from skewness.
Use truncated inverse-power series with coefficients b_k(beta).
Valid for x ≠ 0; typically used on x > 0 and mirrored for x < 0.
- aub_htp.pdf.skorohod.skorohod_formula_2_bk(beta, k)#
Helper for Skorohod’s Formula 2 (alpha = 1): compute b_k(beta).
Integrates e^{-v} v^k * Im[ 1j + beta*1j - (2*beta/π) ln v ] dv over v∈[0, ∞).
Caches results by (beta, k) to speed up repeated calls.
- aub_htp.pdf.skorohod.skorohod_formula_3(x, alpha, beta, N=20)#
Skorohod’s Formula 3: series for 1 < alpha < 2 (tails).
Uses a_n(alpha, beta) with N terms.
Typically applied to |x| large.
- aub_htp.pdf.skorohod.skorohod_formula_3_an(alpha, beta, n)#
Coefficient a_n(alpha, beta) for Skorohod’s Formula 3 (1 < alpha < 2 tails).
Returns array for vector n.
a_n = (-1)^{n-1} (1 + β^2 tan^2(πα/2))^{n/2} sin(n(πα/2 + atan(β tan(πα/2)))) Γ(nα+1)/n!
- aub_htp.pdf.skorohod.skorohod_formula_4(x, alpha)#
Skorohod’s Formula 4: small-|x| asymptotic for 0 < alpha < 1.
Used when x→0 for extreme skew β=±1 handling.
Returns the core exponential form A x^{-1-λ/2} exp(-B x^{-λ}).
- aub_htp.pdf.skorohod.skorohod_formula_4_A(alpha)#
Asymptotic prefactor A(alpha) for 0 < alpha < 1 near zero (β = ±1 edges).
- aub_htp.pdf.skorohod.skorohod_formula_4_B(alpha)#
Asymptotic exponent scale B(alpha) for 0 < alpha < 1 near zero.
- aub_htp.pdf.skorohod.skorohod_formula_4_Lambda(alpha)#
Lambda(alpha) = alpha / (1 - alpha) for 0 < alpha < 1.
- aub_htp.pdf.skorohod.skorohod_formula_5(x)#
Skorohod’s Formula 5: alpha = 1, near zero correction (heuristic).
Empirical correction term to stabilize behavior around 0.
Not a strict series; acts as a patch for numerical issues.
- aub_htp.pdf.skorohod.skorohod_formula_6(x, alpha)#
Skorohod’s Formula 6: small-|x| asymptotic for alpha > 1.
Used when x→0 for extreme skew β=±1 handling.
Returns A’ x^{-1+λ’/2} exp(-B’ x^{λ’}).
- aub_htp.pdf.skorohod.skorohod_formula_6_A_prime(alpha)#
Asymptotic prefactor A’(alpha) for alpha > 1 near zero (β = ±1 edges).
- aub_htp.pdf.skorohod.skorohod_formula_6_B_prime(alpha)#
Asymptotic exponent scale B’(alpha) for alpha > 1 near zero.
- aub_htp.pdf.skorohod.skorohod_formula_6_lambda_prime(alpha)#
Lambda’(alpha) = alpha / (alpha - 1) for alpha > 1.