A block cipher with independent keys for each round I want to modify a standard block cipher in the following way. I replace each round key by a key picked at random. Is this block cipher as secure as the original one ?

Thank you.

EDIT

Some missing information related to the question, according to comment by OP to one of the answers: 


  My question is related to the key-schedule of AES, I wonder if we can replace it by something more stringent and be sure that the security will not be not worst than the original one.
John Kelsey, Bruce Schneier and David Wagner proposed paper "Key-Schedule Cryptanalysis of IDEA, G-DES, GOST, SAFER, and Triple-DES" and they presented new attacks on key schedules of the block ciphers. About "A 768-bit DES variant uses independent round subkeys" they said:  


  A 768-bit DES variant uses independent round subkeys [Ber83]. This
  variant will be much weaker in some situations: there is a very simple
  related-key attack needing just 15 related keys and 60 chosen
  plaintexts. Obtain the encryptions $E(k, p)$ and $E(k' , p)$, where k is
  obtained from k by flipping some bits in the last round subkey; this
  can be thought of as a differential $1R$ attack with a characteristic of
  probability 1. The last round subkey can be recovered with four chosen
  plaintexts, and then we can peel off the last round and repeat the
  attack on 15-round DES. This attack can also be optimized for the case
  when related key queries are very expensive to achieve a complexity of
  one related key and $2^{16}$ or so chosen plaintexts. For nearly any
  product block cipher, if it’s possible to flip bits in a cipher’s
  expanded key, it’s possible to mount an XOR differential attack on the
  last round of the cipher. This may be useful in attacking some systems
  that leave expanded keys vulnerable to change.


So if the algorithm is DES the answer is: NO.  

Charles Bouillaguet, Patrick Derbez, Orr Dunkelman, Nathan Keller and Pierre-Alain Fouque proposed "Low Data Complexity Attacks on AES". They said:

We present several attacks on up to four rounds of AES requiring up to ten chosen plaintexts. Most of the attacks are based on the meet-in-the-middle approach. Some of the attacks exploit heavily the AES key schedule, while others apply even if the subkeys in AES are replaced by independent subkeys. The attacks are summarized in Table 1.

The attacks are applicable to up to 7-rounds of Rijndael. Since the minimal number of rounds in the Rijndael parameter settings proposed for AES is 10, these attacks does not endanger the security of the cipher.  

By now there isn't any attacks against full-rounds of Rijndael key schedule or full-rounds of Rijndael with independent subkeys.  

So if the algorithm is AES the answer is: By now YES
But the answer is depending on future researches. And in general the answer is depending on the algorithm.