Assume that we have a sequence of n realizations x 1 , x 2 , . . . , x n of an i.i.d random variable X with cdf F X ( x ) and pdf f X ( x ) . Now define Y as the sum of k consecutive realizations of X Y i = ∑ j = i k x j which generates the sequence Y 1 , . . . , Y n − k + 1 where Y i contains k − 1 elements of Y i − 1 (and Y i + 1 ).What is the disribution F Y ( x ) of Y?

Question

Assume that we have a sequence of n realizations       x    1    ,      x    2    ,  .  .  .  ,      x    n   of an i.i.d random variable X with cdf       F    X    (  x  ) and pdf       f    X    (  x  ) . Now define Y as the sum of k consecutive realizations of X      Y    i    =      ∑          j      =      i              k            x    j  which generates the sequence       Y    1    ,  .  .  .  ,      Y          n      −      k      +      1       where       Y    i   contains   k  −  1 elements of       Y          i      −      1       (and       Y          i      +      1       ).What is the disribution       F    Y    (  x  ) of Y?

Brogan Navarro · Accepted Answer

Step 1I will denote       Y    k    =      ∑          j      =      i        k        X    j  , as the dependence on k is what matters.As you said, the pdf       f                  Y        k              (  x  ) is given by the k-fold convolution of       f                  Y        k              (  x  ) . You can prove this using the characteristic function of       Y    k   ,       ϕ                  Y        k              (  t  )  =  ⟨      e          i      t              Y        k                  ⟩                  Y        k              =  ⟨      e          i      t              ∑                  j          =          i                k                    X        j                  ⟩                  X        i            ,      …      ,              X                  k          +          i                    You don&#039;t need to distinguish these two cases, as the pdf of       Y    k   depends only on k (not on n). In fact, you can prove that the distribution of       Y    k   does not depend on i. For large k, you have a central limit theorem for       Y    k   .That is true also for finite n

Distribution of sum of i.i.d random variables

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