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Distributions (polykin.distributions)¤

convolve_moments_self ¤

convolve_moments_self(
    q: FloatVectorLike, order: int
) -> FloatVector

Compute the moments of the k-th order convolution of a distribution with itself.

If \(P^{(k)}\) is the \(k\)-th order convolution of \(Q\) with itself, defined as:

\[\begin{aligned} P^{(1)}_n &= Q*Q = \sum_{m=0}^{n} Q_{n-m} Q_{m} \\ P^{(2)}_n &= (Q*Q)*Q = \sum_{m=0}^{n} Q_{n-m} P^{(1)}_{m} \\ P^{(3)}_n &= ((Q*Q)*Q)*Q = \sum_{m=0}^{n} Q_{n-m} P^{(2)}_{m} \\ ... \end{aligned}\]

then the moments of \(P^{(k)}\) are related to the moments of \(Q\) by:

\[\begin{aligned} p_0 &= q_0^{k+1} \\ p_1 &= (k+1) q_0^k q_1 \\ p_2 &= (k+1) q_0^{k-1} (k q_1^2 +q_0 q_2) \\ \ldots \end{aligned}\]

where \(p_i\) and \(q_i\) denote the \(i\)-th moments of \(P^{(k)}\) and \(Q\), respectively.

PARAMETER DESCRIPTION
q

Moments of \(Q\), denoted \((q_0, q_1, \ldots)\).

TYPE: FloatVectorLike(N)

order

Order of the convolution.

TYPE: int

RETURNS DESCRIPTION
FloatVector(N)

Moments of \(P^{(k)}=(Q*Q)*...\), denoted \((p_0, p_1, \ldots)\).

Examples:

>>> from polykin.distributions import convolve_moments_self
>>> convolve_moments_self([1e0, 1e2, 2e4], 2)
array([1.0e+00, 3.0e+02, 1.2e+05])
Source code in src/polykin/distributions/misc.py 
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def convolve_moments_self(
    q: FloatVectorLike,
    order: int
) -> FloatVector:
    r"""Compute the moments of the k-th order convolution of a distribution
    with itself.

    If $P^{(k)}$ is the $k$-th order convolution of $Q$ with itself, defined as:

    \begin{aligned}
    P^{(1)}_n &= Q*Q = \sum_{m=0}^{n} Q_{n-m} Q_{m} \\
    P^{(2)}_n &= (Q*Q)*Q = \sum_{m=0}^{n} Q_{n-m} P^{(1)}_{m} \\
    P^{(3)}_n &= ((Q*Q)*Q)*Q = \sum_{m=0}^{n} Q_{n-m} P^{(2)}_{m} \\
    ...
    \end{aligned}

    then the moments of $P^{(k)}$ are related to the moments of $Q$ by:

    \begin{aligned}
    p_0 &= q_0^{k+1}  \\
    p_1 &= (k+1) q_0^k q_1 \\
    p_2 &= (k+1) q_0^{k-1} (k q_1^2 +q_0 q_2) \\
    \ldots
    \end{aligned}

    where $p_i$ and $q_i$ denote the $i$-th moments of $P^{(k)}$ and $Q$,
    respectively.    

    Parameters
    ----------
    q : FloatVectorLike (N)
        Moments of $Q$, denoted $(q_0, q_1, \ldots)$.
    order : int
        Order of the convolution.

    Returns
    -------
    FloatVector (N)
        Moments of $P^{(k)}=(Q*Q)*...$, denoted $(p_0, p_1, \ldots)$.

    Examples
    --------
    >>> from polykin.distributions import convolve_moments_self
    >>> convolve_moments_self([1e0, 1e2, 2e4], 2)
    array([1.0e+00, 3.0e+02, 1.2e+05])
    """

    q = np.asarray(q, dtype=float)

    if order == 0:
        return q.copy()

    N = len(q)
    if N <= 3:
        # Use closed-form expressions for the first three moments
        p = np.empty(N)
        if N > 0:
            p[0] = q[0]**(order + 1)
        if N > 1:
            p[1] = (order + 1) * q[0]**order * q[1]
        if N > 2:
            p[2] = (order + 1) * q[0]**(order - 1) * (order*q[1]**2 + q[0]*q[2])
    else:
        p = q.copy()
        for _ in range(order):
            p = convolve_moments(q, p)

    return p