Re: Moments of stochastic processes

From: Herman Rubin (hrubin_at_stat.purdue.edu)
Date: 03/02/05

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    In article <cvv856$abn$1@news.ks.uiuc.edu>,
    Michael J Hardy <mjhardy@mit.edu> wrote:
    >I know a roundabout way of proving this proposition.
    >What is the short, straightforward, and easy way that
    >I'm to dumb to see?

    >Let { X_t : t >or= 0 } be a real-valued continuous-time
    >stochastic process with stationary independent increments.
    >Then E(X_t^n) is an nth-degree polynomial in t.

    We need that the increments are independent of
    X_0 and that the n-th moment of an increment
    exists, as well as the n-th moment of X_0. It
    is sufficient to prove that for t=jq/k the
    result holds.

    So look at X_{jq/k} = X_0 + \sum_1^j (X_{iq/k} - X_{(i-1)q/k}.
    All the terms in the sum are independent and identically
    distributed. Now expand the n-th moment. 

    -- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
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