Product integral

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About the product integral

The product integral is a mathematical operator which fills the gap in the following table:

and looks like


This notation was suggested by Gill & Johansen (1990). Although the product integral is relatively unknown to most people, it performs an important role in the theory of survival analysis and Markov processes. In fact, for those people who are familiar with survival analysis, the Kaplan-Meier estimator of the survival function is the product integral of the Nelson-Aalen estimator of the cumulative intensity function. Product integration was introduced by the Italian mathematician Vito Volterra in relation to the Volterra integral equations.


  1. download the LaTeX package from Richard Gill's website:
  2. unpack the archive in an appropriate location of your TeX installation and run texhash to update the filename database
  3. make sure that pdftex, dvips and others can find the font mapfile (e.g. by running updmap --enable if you have teTeX or TeXlive on Linux)
  4. place the following code before the body of your document to add the symbol to the set of mathematical symbols:
\definefontsynonym [MathGamma]  [prodint]

\definefamilysynonym [default]  [xop]   [mc]

\startmathcollection [default]

\definemathsymbol [prodi]  [op]  [xop] [80]
\definemathsymbol [Prodi]  [op]  [xop] [82]
\definemathsymbol [PRODI]  [op]  [xop] [84]



\starttypescript [math] [modern,computer-modern,latin-modern,ams] [size]
[17.3pt,14.4pt,12pt,11pt,10pt,9pt,8pt,7pt,6pt,5pt,4pt] [mm] [mc=prodint]

\definetypeface [modern] [mm] [math]  [modern]




Your text here



The following is the example which comes with the prodint package:

internal error: copy error sfLtTn/cropped.pdf


Please consider the following notes:

  • the installation instructions above are valid when using the Computer Modern font. If you use any other font as bodyfont, the setup should be changed accordingly.
  • this setup defines three commands: \prodi for inline formulae, \Prodi and \PRODI for displaystyle formulae (where the latter is slightly larger); see the example above for demonstration of the usage of these three commands.

References and resources

  • Gill, R.D. & Johansen, S. (1990) A survey of product-integration with a view toward application in survival analysis, The Annals of Statistics, Vol. 18, pp.1501-1555.
  • Richard Gill's website contains links to various articles on product integration