Probability density estimation from sampled data

For broad classes of deterministic and random sampling schemes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{tₖ\}</tex> we establish the consistency and asymptotic expressions for the bias and covariance of discrete-time estimates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f̂ₙ(x)</tex> for the marginal probability density function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x)</tex> of continuous-time processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(t)</tex> . The effect of the sampling scheme and the sampling rate on the performance of the estimates is studied. The results are established for continuous-time processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(t)</tex> satisfying various asymptotic independence-uncorrelatedness conditions.