The k-Record Values of Discrete Random

Variables

Subsequences Xij

X1

X3

X2

X13X2

1

X34

• Where Xi j is the “ith” subsequence and “jth” term of subseqence

K- record value- An entry i is a k-record value if there are exactly k entries ≥ i occurring before i in the

sequence- K-record values exist iff Xi

k = i

- Hence in the 2nd subsequence, 2 is a “one-record value” - In the 3rd subsequence, 3 is a “five-record value”

X13X2

1

X35

Ignatov’s Theorem -> Rk

• Ordered set of record values are independently distributed • Not dependent on the sequence they occur in

X13

X211X3

5

X19 X1

14X21

X214

PMFs of k-record values

Since the Xi k = i’s are all independent and the above PMFs are independent

of the k-record number then Rk values are also independent

Expected entires till first k-record index

We have to find E[T]

Define the first k-record entry as follows:

Consider a sequence of random variables X1, X2, . . . which take on one of the values i, i + 1, . . . ,m with probabilities p1, p2,…. respectively

Since i is the smallest value in the sequence, we check Xk for i

if Xk=i : : where Ti is the first k-record in this sequence

Otherwise we choose subsequence i+1 , where probability of each entry from i+1,..,m is given as :

In this case, if Ti+1 is the number of entries till first k–record entry , we can show that Ti is given as

Where, Ni is the number of occurrences of i in the original sequence, till we reach the Ti+1 entry.

Let Ti+1 be a value n (≥k) , then the distribution of Time taken to get n successes if geometric with the following success probability

Since the mean of the geometric random variable will be 1/(1/ λi) , we get

Taking expectation on both sides

Thus we get the following expectation of T by conditioning on Xk= i and Xk>i