ABSTRACT
One has sometimes to deal with situations where the number of observations is random. For instance, the length of the longest head run (LLHR) in a sequence of random tails and heads (0’s and 1’s) is a maximum of a random number of r.v.s. LLHR (3.2) has interesting applications ranging from reliability theory to psychology and finance; it has been intensively studied by many distinguished authors starting with [235]. Related is the problem of the length of the longest match pattern, which has applications in biology (see [118, 133, 230, 347] and references therein). In this chapter we present results on the asymptotic distribution of the maximum of a random number of random variables and the number of exceedances in samples of random size.
Suppose we observe a sequence {X1, . . . , Xν} of random variables (r.v.s), where ν is a random variables instead of r.v.s integer. One can be interested in the distribution of the sample maximum
