We will use the probabilistic method. Assume
are i.i.d. . Simple algebra yields the equivalency
Denote . It is possible to
show. Moreover, it
holds that any
distributed random variable is sub-Gaussian with
parameter , implying
which gives an upper bound on the probability of (a) being violated.
Moving to the condition (c), we use the result of Example 6.21 from the book.
In particular, we see that for the covariance from the example
and therefore the inequality in (c) reduces to
, i.e.,
the probability of (c) being violated is upper bounded by
Putting everything together, the probability of conditions (a) and (c) being
satisfied is lower bounded by
where we substituted . One can check that the r.h.s. is strictly
larger than zero for and all .