Distributions (polykin.distributions)

convolve_moments_self

convolve_moments_self(
    q0: float, q1: float, q2: float, order: int = 1
) -> tuple[float, float, float]

Compute the first three 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:

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

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

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

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

PARAMETER	DESCRIPTION
q0	0-th moment of \(Q\). TYPE: float
q1	1-st moment of \(Q\). TYPE: float
q2	2-nd moment of \(Q\). TYPE: float

RETURNS	DESCRIPTION
tuple[float, float, float]	0-th, 1-st and 2-nd moments of \(P^k=(Q*Q)*...\).

Examples:

>>> from polykin.distributions import convolve_moments_self
>>> convolve_moments_self(1., 1e2, 2e4, 2)
(1.0, 300.0, 120000.0)

Source code in src/polykin/distributions/base.py

def convolve_moments_self(q0: float,
                          q1: float,
                          q2: float,
                          order: int = 1
                          ) -> tuple[float, float, float]:
    r"""Compute the first three 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_{i=0}^{n} Q_{n-i} Q_{i} \\
    P^2_n &= (Q*Q)*Q = \sum_{i=0}^{n} Q_{n-i} P^1_{i} \\
    P^3_n &= ((Q*Q)*Q)*Q = \sum_{i=0}^{n} Q_{n-i} P^2_{i} \\
    ...
    \end{aligned}

    then the first three 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)
    \end{aligned}

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

    Parameters
    ----------
    q0 : float
        0-th moment of $Q$.
    q1 : float
        1-st moment of $Q$.
    q2 : float
        2-nd moment of $Q$.

    Returns
    -------
    tuple[float, float, float]
        0-th, 1-st and 2-nd moments of $P^k=(Q*Q)*...$.

    Examples
    --------
    >>> from polykin.distributions import convolve_moments_self
    >>> convolve_moments_self(1., 1e2, 2e4, 2)
    (1.0, 300.0, 120000.0)
    """
    p0 = q0**(order+1)
    p1 = (order+1) * q0**order * q1
    p2 = (order+1) * q0**(order-1) * (order*q1**2 + q0*q2)
    return p0, p1, p2