The average of a sequence of random variables does not converge to the mean of each variable in the sequence almost surely, where it is determined that the sequence of averages ( S n ) n = 2 ∞ , S n = 1 n − 1 ∑ m = 2 n X m , P ( X m = m ) = P ( X m = − m ) = 1 2 m log ⁡ ( m ) , P ( X m = 0 ) = 1 − 1 m log ⁡ ( m ) does not converge almost surely to 0. But what do we know about the general behaviour of the said sequence ( S n ) = 2 ∞ ? How could we determine whether or not the sequence converges almost surely to any constant c ∈ R ?

Question

The average of a sequence of random variables does not converge to the mean of each variable in the sequence almost surely, where it is determined that the sequence of averages  (      S    n        )          n      =      2        ∞    ,      S    n    =      1          n      −      1            ∑          m      =      2        n        X    m  ,   P  (      X    m    =  m  )  =  P  (      X    m    =  −  m  )  =      1          2      m      log      ⁡      (      m      )        ,  P  (      X    m    =  0  )  =  1  −      1          m      log      ⁡      (      m      )       does not converge almost surely to 0. But what do we know about the general behaviour of the said sequence   (      S    n        )          =      2        ∞  ? How could we determine whether or not the sequence converges almost surely to any constant   c  ∈      R  ?

Rubi Boyle · Accepted Answer

Suppose       S    n   converges to a constant   c  ≠  0? Using Chebychev&#039;s Inequality:  Pr  (      |        S    n        |    &amp;gt;  λ  )  ≤      λ          −      2            E        |        S    n              |        2    .Now      E        |        S    n              |        2    =      1          (      n      −      1              )        2                  ∑          m      =      2        n              m      2              m      log      ⁡      (      m      )        ,which is bounded by   C      /    log  ⁡  (  n  ) for some constant   C  &amp;gt;  0 if   n  ≥  2, because                              ∑                      m            =            2                    n                                      m            2                                m            log            ⁡            (            m            )                                              ≤                  ∑                      m            =            2                                ⌊                          n                        ⌉                                    m                      log            ⁡            (            m            )                          +                  ∑                      m            =            ⌈                          n                        ⌉                    n                          m                      log            ⁡            (            m            )                                                            ≤                  ∑                      m            =            2                                ⌊                          n                        ⌉                          m        +                  ∑                      m            =            ⌈                          n                        ⌉                    n                          m                      log            ⁡            (                          n                        )                                                            ≤                  ∑                      m            =            2                                ⌊                          n                        ⌉                                    n                +                  ∑                      m            =            1                    n                          n                      log            ⁡            (                          n                        )                                                            ≤        n        +        2                              n            2                                log            ⁡            (            n            )                                                            ≤                              n            2                                log            ⁡            (            n            )                          +        2                              n            2                                log            ⁡            (            n            )                                                            ≤        12                              (            n            −            1                          )              2                                            log            ⁡            (            n            )                          ,            since   log  ⁡  (  n  )  ≤  n, and       n    2    ≤  4  (  n  −  1      )    2  .So   Pr  (      |        S    n        |    &amp;gt;      |    c      |        /    2  )  →  0 as   n  →  ∞.

The average of a sequence of random variables does not converge to the mean of each variable in the

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